1 Introduction
Work in
$ \mathsf {ZF} + \mathsf {DC}(\mathbb {R}) $
.Footnote
1
Given topological spaces X and Y, a continuous reduction from
$A\subseteq X$
to
$B\subseteq Y$
is a continuous function
$f \colon X\to Y$
satisfying
$f^{-1}(B) = A$
. When such an
$ f $
exists we say that
$A\subseteq X$
continuously reduces or Wadge reduces to
$B\subseteq Y$
, and we write
$A\le _{\mathsf {W}}^{X,Y}B$
, or
$A\le _{\mathsf {W}}^{X}B$
if
$X=Y$
. Notice that
$ A \le _{\mathsf {W}}^{X,Y} B $
if and only if
$ {X \setminus A} \le _{\mathsf {W}}^{X,Y} {X \setminus B} $
. We also denote by
$ <_{\mathsf {W}}^{X ,Y}$
the strict part of
$ \le _{\mathsf {W}}^{X,Y} $
, that is:
$ A <_{\mathsf {W}}^{X,Y} B \iff { {A \le _{\mathsf {W}}^{X,Y} B} \wedge {B \not \le _{\mathsf {W}}^{Y,X} A}}$
. Similarly, we write
$ A \equiv _{\mathsf {W}}^{X,Y} B \iff {{A \le _{\mathsf {W}}^{X,Y} B} \wedge {B \le _{\mathsf {W}}^{Y,X} A}} $
. To simplify the notation, we again write
$ <_{\mathsf {W}}^X $
and
$ \equiv _{\mathsf {W}}^X $
instead of
$ <_{\mathsf {W}}^{X,X} $
and
$ \equiv _{\mathsf {W}}^{X,X} $
, respectively.
Continuous reducibility is a transitive and reflexive relation, that is a quasi-order, and
$ \equiv _{\mathsf {W}}^X $
is the equivalence relation canonically induced by
$ \le _{\mathsf {W}}^X $
. We call Wadge degree of
$A\subseteq X$
the
$\equiv _{\mathsf {W}}^{X}$
-equivalence class of A, and we denote it by
$[A]_{\mathsf {W}}^X$
. The Wadge reducibility
$ \le _{\mathsf {W}}^X $
induces a partial order on Wadge degrees of X that we call Wadge hierarchy on X and denote by
. When
$ \le _{\mathsf {W}}^X $
is well-founded, we can associate with each set
$ A \subseteq X $
(or to its Wadge degree
$ [A]_{\mathsf {W}}^{X} $
) its rank according to
$ \le _{\mathsf {W}}^{X} $
, which is called the Wadge rank of
$ A $
(or of
$ [A]_{\mathsf {W}}^X $
) and is denoted by
$ ||A||_{\mathsf {W}}^{X} $
. For technical and historical reasons, we use
$ 1 $
as the minimal value for
$ ||A||_{\mathsf {W}}^{X} $
. We also let
$ \Theta _X $
be the length of
, i.e.,
$ \Theta _X = \sup \{ ||A||_{\mathsf {W}}^X +1 \mid A \subseteq X \} $
.
To simplify the notation, given a set
$ A \subseteq X $
we sometimes write
$ \neg A $
instead of
$ X \setminus A $
when the ambient space
$ X $
is clear from the context. Wadge left his name to continuous reducibility when he proved in [Reference Wadge33], by game theoretic methods, that
$\le _{\mathsf {W}}^{\mathcal {N}}$
on Borel subsets of the Baire space
$\mathcal {N} = {}^\omega \omega $
is well-founded and satisfies the Semi-Linear Ordering principle, or
$ \mathsf {SLO}^{\mathsf {W}}_{\mathcal {N}} (\mathrm {Bor}) $
: for all (Borel) sets
$A,B \subseteq \mathcal {N}$
$$ \begin{align} A \le_{\mathsf{W}}^{\mathcal{N}} B \quad \text{or} \quad \neg B \le_{\mathsf{W}}^{\mathcal{N}} A. \end{align} $$
These two results generalize under the Axiom of Determinacy (
$\mathsf {AD}$
), yielding to the fact that the Wadge hierarchy is semi-well-ordered, namely: the full
$ \mathsf {SLO}^{\mathsf {W}}_{\mathcal {N}} $
holds (i.e., (1.1) is true for arbitrary subsets
$ A,B \subseteq \mathcal {N} $
), and
$ \le _{\mathsf {W}}^{\mathcal {N}} $
is well-founded. (The former directly follows from Wadge’s original lemma, while the latter is due to a highly nontrivial argument by Martin and Monk.)
Early investigations on Wadge theory [Reference Solovay, Kechris and Moschovakis29–Reference Van Wesep31, Reference Wadge33] have fully described the Wadge hierarchy on
$ \mathcal {N}$
under
$\mathsf {AD}$
(see Figure 1). Indeed, by
$ \mathsf {SLO}^{\mathsf {W}}_{\mathcal {N}} $
one easily gets that for each
$ A \subseteq \mathcal {N} $
, either
$ A \le _{\mathsf {W}}^{\mathcal {N}} \neg A $
, in which case we say that
$ A $
(or its Wadge degree
$ [A]_{\mathsf {W}}^{\mathcal {N}} $
) is selfdual, or else
$ A $
and
$ \neg A $
are
$ \le _{\mathsf {W}}^{\mathcal {N}} $
-incomparable, in which case
$ A $
is called nonselfdual and
$ \{ [A]_{\mathsf {W}}^{\mathcal {N}}, [\neg A]_{\mathsf {W}}^{\mathcal {N}} \} $
is a maximal antichain, called a nonselfdual pair. The minimal Wadge degrees are those in the nonselfdual pair
$ \{ [\mathcal {N}]_{\mathsf {W}}^{\mathcal {N}}, [\emptyset ]_{\mathsf {W}}^{\mathcal {N}} \} $
, and indeed
$ [\mathcal {N}]_{\mathsf {W}}^{\mathcal {N}} = \{ \mathcal {N} \} $
and
$ [\emptyset ]_{\mathsf {W}}^{\mathcal {N}} = \{ \emptyset \} $
. After this minimal nonselfdual pair there is a single selfdual degree consisting of all nontrivial clopen subsets of
$ \mathcal {N} $
. Then nonselfdual pairs and selfdual degrees alternate, at limit levels of countable cofinality there is always a selfdual degree, whereas at limit levels of uncountable cofinality there is always a nonselfdual pair. Finally,
$ \Theta _{\mathcal {N}} = \Theta $
where
$$\begin{align*}\Theta = \sup \{ \alpha \in \mathrm{Ord} \mid \mathbb{R} \text{ surjects onto } \alpha \}. \end{align*}$$
Figure 1 The Wadge hierarchy
on the Baire space
$ \mathcal {N} $
.
Wadge theory has received a lot of attention, as witnessed by [Reference Andretta3, Reference Andretta and Martin5, Reference Carroy, Medini and Müller10, Reference Duparc12, Reference Kihara and Montalbán18–Reference Motto Ros23, Reference Van Wesep, Kechris and Moschovakis32], among many other works. Several generalizations have been considered, including variations of the reducibility (usually replacing continuous functions with other natural classes of functions) and/or replacing
$ \mathcal {N} $
with other spaces. In the second direction, the first natural move is to consider other zero-dimensional Polish spaces.Footnote
2
For example, if one assumes
$ \mathsf {AD}$
again and considers the Cantor space
$ \mathcal {C} = {}^\omega 2 $
, then the corresponding Wadge hierarchy
looks exactly like
except for the fact that at limit levels one always has a nonselfdual pair, independently of the cofinality of the level (see Figure 2). This is obtained via straightforward modifications of the results and game-theoretic methods used in the case of
$ \mathcal {N} $
.Footnote
3
Figure 2 The Wadge hierarchy
on the Cantor space
$ \mathcal {C} $
.
If one moves to an arbitrary zero-dimensional Polish space
$ X $
(equivalently: to an arbitrary closed subspace of
$ \mathcal {N} $
) different from
$ \mathcal {N} $
and
$ \mathcal {C} $
, instead, very little is known about how
looks like. The hierarchy starts again with the nonselfdual pair
$ \{ [X]_{\mathsf {W}}^X, [\emptyset ]_{\mathsf {W}}^X \} $
, where
$ [X]_{\mathsf {W}}^X = \{ X \} $
and
$ [\emptyset ]_{\mathsf {W}}^X = \{ \emptyset \} $
, followed by the selfdual degree of all nontrivial clopen subsets of
$ X $
. As discussed in Section 2.2, the existence of a retraction from
$ \mathcal {N} $
onto
$ X $
yields the fact that, still under
$ \mathsf {AD} $
, the Wadge hierarchy
is semi-well-ordered, i.e., the quasi-order
$ \le _{\mathsf {W}}^X $
is well-founded and
$ \mathsf {SLO}^{\mathsf {W}}_X $
is true, i.e.,
$ {A \le _{\mathsf {W}}^X B} \vee {\neg B \le _{\mathsf {W}}^X A}$
holds for all
$ A,B \subseteq X $
(see also [Reference Motto Ros, Schlicht and Selivanov24]). This implies that at each level of
we find either a nonselfdual pair
$ \{ [A]_{\mathsf {W}}^X, [\neg A]_{\mathsf {W}}^X \} $
(when
$ A \not \le _{\mathsf {W}}^X \neg A $
) or a single selfdual degree
$ [A]_{\mathsf {W}}^X $
(when
$ A \le _{\mathsf {W}}^X \neg A $
). Another easy observation is that
$ \Theta _X = \Theta _{\mathcal {N}} = \Theta $
if
$ X $
is uncountable, while
$ \Theta _X < \omega _1 $
if
$ X $
is countable. This follows from the retraction trick again (see Section 2.2) if
$ X $
is uncountable, because in this case
$ X $
contains a closed homeomorphic copy of
$ \mathcal {C} $
; if instead
$ X $
is countable, we use the fact that every subset of
$ X $
is a
$ \boldsymbol {\Delta }^0_2 $
-set, and indeed it is in
$ D_{\alpha }(\boldsymbol {\Sigma }^0_1) $
, where
$ 1 \leq \alpha < \omega _1 $
depends on the Cantor–Bendixson rank
$ ||X||_{\operatorname {CB}} $
of
$ X $
. However, nothing else is known, and a complete and detailed description of
has never been carried out in the literature: the main purpose of this article is to fill this gap and prove the following result. (Notice that in the countable case zero-dimensionality is automatic and could thus be removed from the hypotheses, and moreover in such case we do not need any determinacy assumptions because
and thus Martin’s Borel determinacy is enough.)
Main Theorem 1. Let
$ X $
be an arbitrary zero-dimensional Polish space, and assume
$ \mathsf {AD} $
if
$ X $
is uncountable. Then:
-
(1)
is semi-well-ordered (Proposition 2.3). -
(2) Selfdual degrees and nonselfdual pairs alternate, starting with the nonselfdual pair
$ \{ [X]_{\mathsf {W}}^X, [\emptyset ]_{\mathsf {W}}^X \} $
at the bottom, followed by the selfdual degree of all nontrivial clopen sets (Theorem 4.4, Theorem 4.6, and Proposition 2.2). -
(3) For limit levels, we distinguish two cases:
-
(a) If the perfect kernel of
$ X $
is not compact (hence nonempty), then at all limit levels of uncountable cofinality there is a nonselfdual pair and at all limit levels of countable cofinality there is a selfdual degree (Proposition 4.7 and Theorem 4.9). -
(b) Otherwise there is
$ \alpha _X < \omega _1 $
such that for all limit
$ \alpha < \Theta _X $
, there is a selfdual degree at level
$ \alpha $
if and only if
$ \alpha < \alpha _X ;$
in particular,
$ \alpha _X = 0 $
if
$ X $
is compact (Corollary 4.14 and the ensuing discussion).
-
-
(4)
$ \Theta _X = \Theta $
if
$ X $
is uncountable, while if
$ X $
is countable then
$\Theta _X = 2 \cdot ||X||_{\operatorname {CB}} + \varepsilon _X$
for some
$\varepsilon _X \in \{-1,0,1\}$
(Proposition 2.5 and Theorem 4.15).
The parameter
$ \alpha _X $
depends on a new rank on
$ X $
which we call compact rank and denote by
$ ||X||_{\operatorname {Comp}} $
(Section 3.3). As for item (4), the ordinal
$ ||X||_{\operatorname {CB}} $
is the usual Cantor–Bendixson rank of
$ X $
, while the parameter
$ \varepsilon _X $
depends on
$ ||X||_{\operatorname {CB}} $
,
$ \alpha _X $
, and whether
$ X $
is simple or not (see Section 2.5 for the relevant definitions).
It follows from Main Theorem 1 that if
$ X $
has a non-compact perfect kernel, then
is isomorphic to
(Figure 1); if
$ X $
is compact and, more generally, if
$ ||X||_{\operatorname {Comp}} < \omega $
, then
is isomorphic to
(Figure 2); in the remaining cases, i.e., when
is isomorphic neither to
nor to
, the hierarchy
is a mixture of the two, with an initial segment (of countable length) which is Baire-like and the rest of the hierarchy which is Cantor-like (see Figure 3).
Figure 3 The Wadge hierarchy
on an arbitrary zero-dimensional Polish space
$ X $
when
is isomorphic neither to
nor to
. In this case, the ordinal
$ \alpha _X $
satisfies
$ \omega < \alpha _X < \omega _1 $
.
Main Theorem 1 is optimal, in the sense that for all possibilities which are coherent with the limitations contained therein, there is a zero-dimensional Polish space
$ X $
whose Wadge hierarchy
realizes precisely that scenario (Section 4.4). Thus what we have obtained is a complete classification (up to order-isomorphism) of all possible Wadge hierarchies for zero-dimensional Polish spaces, where the complete invariants are given by pairs of ordinals
$ (\alpha _X, \Theta _X ) $
which can easily be coded as countable ordinals (see the discussion at the beginning of Section 5).
Although our classification of the possible Wadge hierarchies over zero-dimensional Polish spaces
$ X $
is complete and quite satisfactory, we also show that it is not possible to determine
in a Borel manner. For example, by Theorem 5.3 the collection of all zero-dimensional Polish spaces whose Wadge hierarchy is order-isomorphic to
is a complete analytic subset of the Effros Borel space
$ F(\mathcal {N}) $
. More generally, for all nontrivial zero-dimensional Polish spaces
$ X $
there is no Borel procedure to determine which zero-dimensional Polish spaces
$ Y $
give rise to a Wadge hierarchy
order-isomorphic to
.
Main Theorem 2. Let
$ X $
be a zero-dimensional Polish space with at least two points, and assume
$ \mathsf {AD} $
if
$ X $
is uncountable. Then the set of zero-dimensional Polish spaces Y such that
is order-isomorphic to
is not Borel in the Effros Borel space
$ F(\mathcal {N})$
.
Indeed, it is not even possible to decide in a Borel way whether at a given countable limit level we will find a nonselfdual pair or a selfdual degree, or to determine the value of
$ \alpha _X $
and/or
$ \Theta _X $
(see Remark 5.4). The reason behind this wild behaviour actually relies on the possible configurations of isolated points. If we restrict to perfect spaces, then the situation is radically neater and we have only two (Borel) possibilities: spaces with a Baire-like Wadge hierarchy, and spaces with a Cantor-like Wadge hierarchy (Proposition 5.5).
Finally, let us briefly discuss the optimality of the various hypotheses in our results. As it is well-known in the area, the determinacy assumption
$ \mathsf {AD} $
can be relaxed to get level-by-level statements, at the cost of making such statements and the various definitions a lot more cumbersome. For a detailed discussion on this matter, and specifically on nice pointclasses, see [Reference Carroy, Medini and Müller11, Section 3]. In particular, if we restrict the attention to the Borel subsets of
$ X $
, then Martin’s Borel determinacy suffices and we do not need to assume
$ \mathsf {AD} $
anywhere.
Moreover, we point out that there is a reason for restricting to zero-dimensional Polish spaces. The Wadge quasi-order has been studied in various other contexts, even beyond the realm of Polish spaces (see, for instance, [Reference Camerlo6, Reference Camerlo and Massaza8, Reference Ikegami, Schlicht and Tanaka15, Reference Motto Ros, Schlicht and Selivanov24, Reference Pequignot25, Reference Selivanov28]), but in most cases it is not semi-well-ordered if the space is not zero-dimensional (with a few exceptions: see [Reference Camerlo and Massaza8, Theorem 33]). If one restricts the attention to Polish spaces, this becomes even a characterization because
has infinite antichains whenever
$ X $
is a metric (not necessarily complete or separable) space with positive dimension (see [Reference Schlicht26, Theorem 1.5]). Moreover, [Reference Hertling14] and [Reference Duparc and Vuilleumier13] show among other things that
is ill-founded when
$ X $
is
$\mathbb {R}$
or the Scott domain, respectively. See [Reference Motto Ros, Schlicht and Selivanov24] for more possible behaviours and a rough classification of Wadge-like hierarchies on arbitrary (quasi-)Polish spaces.
2 Preliminaries and known facts
The monograph [Reference Kechris17] is our reference for all classical results and notation in descriptive set theory. From this point onward, unless otherwise stated, we assume
$ \mathsf {AD} $
and that
$ X $
,
$ Y $
,
$ Z $
, … are nonempty zero-dimensional Polish spaces. Recall however that the determinacy assumption is just for ease of exposition, and that no further assumption is required if all results and definitions are restricted to the realm of Borel sets.
2.1 Some easy and well-known facts
The principle
$ \mathsf {SLO}^{\mathsf {W}}_X $
can be reformulated to deal with sets in possibly different zero-dimensional Polish spaces [Reference Kechris17, Theorem 21.14]. The proof is identical to the original one by Wadge, except that the rules of the so-called Wadge’s game are modified so that the players stay inside their respective closed subspaces of
$ \mathcal {N} $
.
Proposition 2.1 (
$ \mathsf {AD} $
).
Let
$ X,Y $
be zero-dimensional Polish spaces. Then for all
$A\subseteq X$
and
$B\subseteq Y$
,
$$\begin{align*}A\le_{\mathsf{W}}^{X,Y}B \quad \vee \quad \neg B\le_{\mathsf{W}}^{Y,X} A. \end{align*}$$
Notice that when
$ X = Y $
, Proposition 2.1 is simply the assertion that
$ \mathsf {SLO}^{\mathsf {W}}_X $
holds under
$ \mathsf {AD} $
for an arbitrary zero-dimensional Polish space
$ X $
. Proposition 2.1 is so crucial in the theory that we will simply refer to it as Wadge’s Lemma.
Let
$ \mathrm {Set}_{\operatorname {SD}}(X) $
be the collection of all selfdual sets of
$ X $
, i.e.,
$$\begin{align*}\mathrm{Set}_{\operatorname{SD}}(X) = \{ A \subseteq X \mid A \le_{\mathsf{W}}^X \neg A \}, \end{align*}$$
and, dually, let
$ \mathrm {Set}_{\operatorname {NSD}}(X) $
be the collection of nonselfdual sets of
$ X $
, that is,
Then Wadge’s Lemma implies that at each level of
we find either a
$ 2 $
-sized maximal antichain
$ \{ [A]_{\mathsf {W}}^X, [\neg A]_{\mathsf {W}}^X \} $
for some
$ A \in \mathrm {Set}_{\operatorname {NSD}}(A) $
, called a nonselfdual pair, or a single selfdual degree
$ [A]_{\mathsf {W}}^X $
for some
$ A \in \mathrm {Set}_{\operatorname {SD}}(X) $
.
Another easy observation (see [Reference Camerlo6, Lemma 1]) is that for every nonempty topological space
$ X $
the pair
$ \{ [X]_{\mathsf {W}}^X, [\emptyset ]_{\mathsf {W}}^X \} $
is nonselfdual and its elements are minimal in
; if
$ X $
is not connected then right after such pair there is a single selfdual degree. (The proof of the next proposition is left to the reader.)
Proposition 2.2. Let
$ X $
be an arbitrary nonempty topological space. Then
$ [X]_{\mathsf {W}}^X = \{ X \} $
and
$ [\emptyset ]_{\mathsf {W}}^X = \{ \emptyset \}$
. Moreover, the pair
$ \{ [X]_{\mathsf {W}}^X, [\emptyset ]_{\mathsf {W}}^X \} $
is nonselfdual and
$ \le _{\mathsf {W}}^X $
-minimal in
, that is, for every
$ \emptyset , X \neq A \subseteq X$
we have
$ \emptyset , X <_{\mathsf {W}}^X A $
.
If moreover
$ X $
is disconnected, then immediately above
$ \{ [X]_{\mathsf {W}}^X, [\emptyset ]_{\mathsf {W}}^X \} $
there is a single selfdual degree consisting of all nontrivial clopen subsets of
$ X $
, and all other Wadge degrees are strictly
$ \le _{\mathsf {W}}^X $
-above it.
2.2 The retraction method
Part (1) of Main Theorem 1 can be proved using a trick due to Marcone (unpublished, but see [Reference Andretta3, Proposition 28]) and, independently, to Selivanov [Reference Selivanov28, Corollary 2.4]. If
$ X \subseteq Y $
are such that there is a retractionFootnote
4
$ \rho $
from
$ Y $
onto
$ X $
, then the map from
to
defined by
$ A \mapsto \rho ^{-1}(A) $
embeds the structure
into
(see also [Reference Motto Ros, Schlicht and Selivanov24, Proposition 5.4] for a more general result). Notice that up to Wadge equivalence, the choice of
$ \rho $
is irrelevant: if
$ \rho _1, \rho _2 \colon Y \to X $
are two retractions then
$ \rho _1^{-1}(A) \equiv _{\mathsf {W}}^Y \rho _2^{-1}(A) $
, as witnessed by the maps
$ \rho _1 $
and
$ \rho _2 $
themselves. Notice also that for any retraction
$ \rho \colon Y \to X $
and
$ A \subseteq X $
, we have
$ A \equiv _{\mathsf {W}}^{X,Y} \rho ^{-1}(A) $
via the inclusion map
$ \iota \colon X \to Y $
and the retraction
$ \rho \colon Y \to X $
.
It is a classical result that every zero-dimensional Polish space can be construed as a closed subspace of
$ \mathcal {N} $
, thus there is a retraction of
$ \mathcal {N} $
onto
$ X $
. Therefore we get the desired result [Reference Motto Ros, Schlicht and Selivanov24, Theorem 5.12(1)].
Proposition 2.3 (
$ \mathsf {AD} $
).
Let
$ X $
be a zero-dimensional Polish space. Then
$ \mathsf {SLO}^{\mathsf {W}}_X $
holds and
$ \le _{\mathsf {W}}^X $
is well-founded, i.e.,
is semi-well-ordered.
Proposition 2.3 allows us to split ordinals
$ 1 \leq \alpha < \Theta _X $
in two groups: those for which at level
$ \alpha $
of
we find a single selfdual degree and those for which we instead find a nonselfdual pair. This is condensed in the following definition and notation.
Definition 2.4. An ordinal
$1 \leq \alpha < \Theta _X$
is called selfdual for X if the subsets of
$ X $
with Wadge rank
$\alpha $
are selfdual, otherwise we say that
$\alpha $
is nonselfdual. We denote by
$ \mathrm {Ord}_{\mathrm {SD}}(X)$
the set of all
$ 1 \leq \alpha < \Theta _X $
which are selfdual for X, while
$ \mathrm {Ord}_{\mathrm {NSD}}(X)$
stands for the set of nonselfdual
$ 1 \leq \alpha < \Theta _X $
.
Notice that by Proposition 2.3, determining the structure of
amounts to determine the two sets of ordinals
$ \mathrm {Ord}_{\mathrm {SD}}(X) $
and
$ \mathrm {Ord}_{\mathrm {NSD}}(X) $
, together with the exact value of
$ \Theta _X $
.
The same trick based on retractions can also be used to prove the first half of part (4) of Main Theorem 1, i.e., to compute
$ \Theta _X $
when
$ X $
is uncountable. Indeed, if a zero-dimensional Polish space
$ X $
is uncountable, then
$ \mathcal {C} $
is (homeomorphic to) a closed subset of
$ X $
: therefore there is a retraction of
$ X $
onto
$ \mathcal {C} $
, and thus
embeds into
. Combined with the fact that, as observed,
embeds into
, this shows that
$ \Theta _{\mathcal {C}} \leq \Theta _X \leq \Theta _{\mathcal {N}} $
. Since under
$ \mathsf {AD} $
it holds
$ \Theta _{\mathcal {C}} = \Theta _{\mathcal {N}} = \Theta $
, we get:
Proposition 2.5 (
$ \mathsf {AD} $
).
Let
$ X $
be an uncountable zero-dimensional Polish space. Then
has length
$ \Theta _X = \Theta $
.
Finally, combining the retraction method with a well-known result in Wadge theory due to Wadge himself [Reference Andretta2, Lemma 3], we obtain a technical fact that provides a useful characterization of selfdual sets.
Proposition 2.6 (
$ \mathsf {AD} $
).
Let
$ X $
be a zero-dimensional Polish space. Then
$ A \subseteq X $
is selfdual if and only if there are a clopen partition
$ (V_n)_{n \in \omega } $
of
$ X $
and sets
$ (A_n)_{n \in \omega } $
such that
$ A_n <_{\mathsf {W}}^X A $
and
$ A = \bigcup _{n \in \omega } (A_n \cap V_n) $
.
Moreover, we can further assume that each
$ A_n $
is nonselfdual, and
$ ||A||_{\mathsf {W}}^X = \sup _{n \in \omega } (||A_n||_{\mathsf {W}}^X+1) $
.
Proof The backward direction is obvious, as by
$ \mathsf {SLO}^{\mathsf {W}}_X$
we have
$ A_n \le _{\mathsf {W}}^X \neg A $
for all
$ n \in \omega $
: if
$ (f_n)_{n \in \omega } $
are continuous maps witnessing this, then
$ f = \bigcup _{n \in \omega } (f_n \restriction V_n) $
is a continuous map witnessing
$ A \le _{\mathsf {W}}^X \neg A $
.
Conversely, let
$ X $
be a closed subspace of
$ \mathcal {N} $
and
$ \rho \colon \mathcal {N} \to X $
be a retraction. If
$ A $
is selfdual in
$ X $
, then
$ A' = \rho ^{-1}(A) $
is selfdual in
$ \mathcal {N} $
. Then the set of those
$ s \in \omega ^{<\omega } $
such that
$ A' \le _{\mathsf {W}}^{\mathcal {N},\boldsymbol {N}_s} A' \cap \boldsymbol {N}_s $
is a nonempty well-founded tree
$ \boldsymbol {T}(A') \subseteq \omega ^{<\omega } $
[Reference Andretta1, Lemma 22]. Let
$ (V_n)_{n \in \omega } $
be an enumeration of the sets of the form
$ \boldsymbol {N}_{s {}^{\smallfrown {}} k} \cap X $
with
$ k \in \omega $
and
$ s $
a leaf in
$ \boldsymbol {T}(A') $
, and set
$ A_n = X $
if
$ V_n \subseteq A $
and
$ A_n = A \cap V_n $
otherwise. It is easy to check that the
$ V_n $
’s and
$ A_n $
’s are as required.
The additional part on nonselfduality of the
$ A_n $
’s follows by iterating the construction on those
$ A_n $
which happen to be selfdual in
$ X $
. (This process must terminate on each branch after finitely many steps because otherwise we would construct a strictly
$ \le _{\mathsf {W}}^X $
-decreasing sequence of subsets of
$ X $
, against Proposition 2.3.)
Finally, set
$ \alpha = ||A||_{\mathsf {W}}^X $
and
$ \alpha _n = ||A_n||_{\mathsf {W}}^X $
, so that in particular
$ \alpha _n < \alpha $
. If
$ B \subseteq X $
is a set with
$ ||B||_{\mathsf {W}}^X = \sup _{n \in \omega } (\alpha _n+1) $
, then
$ A_n <_{\mathsf {W}}^X B $
because
$ ||A_n||_{\mathsf {W}}^X = \alpha _n < \alpha _n+1 \leq ||B||_{\mathsf {W}}^X $
: if
$ f_n $
witnesses
$ A_n \le _{\mathsf {W}}^X B $
, then
$ A \le _{\mathsf {W}}^X B $
via
$ \bigcup _{n \in \omega } (f_n \restriction V_n) $
, thus
$ \sup _{n \in \omega } (\alpha _n+1) = ||B||_{\mathsf {W}}^X \geq ||A||_{\mathsf {W}}^X = \alpha $
. The other inequality is obvious.
Remark 2.7.
-
(1) The above proof shows that we can indeed assume that either
$ A_n = X $
(if
$ V_n \subseteq A $
), or else
$ A_n = A \cap V_n $
(and thus
$ A_n \subseteq V_n $
); indeed, if
$A $
is not clopen and we drop the requirement that
$A_n $
be nonselfdual, then the latter can be assumed to be true for all
$ n \in \omega $
. -
(2) Proposition 2.6 can be relativized to any clopen subset of
$ X $
, namely: If
$A\subseteq X$
is selfdual and
$U\in \boldsymbol {\Delta }^0_1(X)$
, then there exist a clopen partition
$(U_n)_{n \in \omega }$
of
$ U $
and nonselfdual sets
$A_n<_{\mathsf {W}}^X A$
such that
$\bigcup _{n \in \omega } (A_n\cap U_n)=A\cap U$
. Indeed, this is trivial if
$ U \cap A <_{\mathsf {W}}^X A $
. If instead
$ U \cap A \equiv _{\mathsf {W}}^X A $
, then
$ U \cap A $
is selfdual and we can apply Proposition 2.5, setting then
$ U_n = V_n \cap U $
. This easily provides alternative proofs of [Reference Carroy, Medini and Müller11, Theorem 5.4 and Corollary 5.5].
2.3 Pointclasses
Wadge pointclasses provide an alternative (but equivalent) way to present and study the Wadge hierarchy on a space
$ X $
. A boldface pointclass
$ \boldsymbol {\Gamma } $
in X is a subset of
which is closed under continuous preimages, i.e., it is downward closed under
$ \le _{\mathsf {W}}^X $
. The dual
of
$ \boldsymbol {\Gamma } $
is defined by
. The dual operator is clearly idempotent, i.e., the dual of
is
$ \boldsymbol {\Gamma } $
itself. We say that
$ \boldsymbol {\Gamma } $
is nonselfdual if
, and selfdual otherwise.
A boldface pointclass
$ \boldsymbol {\Gamma } $
is a Wadge class if it is of the form
$A {\downarrow }_X=\{ B\subseteq X \mid B\le _{\mathsf {W}}^X A \}$
for some
$ A \subseteq X $
; any
$ A \in \boldsymbol {\Gamma } $
satisfying
$ \boldsymbol {\Gamma } = A {\downarrow }_X $
is called complete (for
$ \boldsymbol {\Gamma } $
), and we say that
$ \boldsymbol {\Gamma } $
is generated by
$ A $
. We denote by
$ \mathrm {Class}_{\mathsf {W}}(X) $
the collection of all Wadge classes in
$ X $
. Notice that by
$ \mathsf {SLO}^{\mathsf {W}}_X $
, every nonselfdual boldface pointclass
$ \boldsymbol {\Gamma } $
of
$ X $
is automatically a Wadge class; in contrast, not all selfdual boldface pointclasses in
$ X $
are Wadge classes. We denote by
$ \mathrm {Class}_{\operatorname {NSD}}(X) $
the collection of all nonselfdual Wadge classes in
$ X $
, and by
$ \mathrm {Class}_{\operatorname {SD}}(X) = \mathrm {Class}_{\mathsf {W}}(X) \setminus \mathrm {Class}_{\operatorname {NSD}}(X) $
the collection of the selfdual ones.
The Wadge hierarchy
is clearly isomorphic to the structure of all Wadge classes ordered by inclusion, as witnessed by the map
$ [A]_{\mathsf {W}}^X \mapsto A {\downarrow }_X $
. In particular,
$ A \in \mathrm {Set}_{\operatorname {SD}}(X) $
if and only if
$ A {\downarrow }_X \in \mathrm {Class}_{\operatorname {SD}}(X) $
, and a nonselfdual pair
$ \{ [A]_{\mathsf {W}}^X, [\neg A]_{\mathsf {W}}^X \} $
corresponds to the pair of distinct nonselfdual Wadge classes
with
$ \boldsymbol {\Gamma } = A {\downarrow }_X $
. Every
$\boldsymbol {\Gamma } \in \mathrm {Class}_{\mathsf {W}}(X)$
induces a coarse Wadge class
. When
$ \mathsf {SLO}^{\mathsf {W}}_X $
holds and
$ \le _{\mathsf {W}}^X $
is well-founded, inclusion on coarse Wadge classes is a well-order: the ordinal
$ \alpha $
corresponding to the position of
$ \boldsymbol {\Gamma }^* $
in such a well-order (once we start counting by
$ 1 $
) coincides with
$ ||A||_{\mathsf {W}}^X $
for some/any
$ A \subseteq X $
such that
$ \boldsymbol {\Gamma } = A {\downarrow }_X $
. Moreover, when
$ X \subseteq Y $
are zero-dimensional Polish spaces with
$ X $
closed in
$ Y $
, the retraction method (Section 2.2) induces an injection from
$ \mathrm {Class}_{\mathsf {W}}(X) $
into
$ \mathrm {Class}_{\mathsf {W}}(Y) $
. Indeed, if
$ \rho \colon Y \to X $
is a retraction we can associate with each
$ \boldsymbol {\Gamma } = A {\downarrow }_X \in \mathrm {Class}_{\mathsf {W}}(X) $
the Wadge class
$ \rho ^{-1}(\boldsymbol {\Gamma }) = (\rho ^{-1}(A)) {\downarrow }_Y \in \mathrm {Class}_{\mathsf {W}}(Y) $
. This is again independent on the chosen retraction, i.e., if
$ \rho _1, \rho _2 \colon Y \to X $
are both retractions, then
$ \rho _1^{-1}(\boldsymbol {\Gamma }) = \rho _2^{-1}(\boldsymbol {\Gamma }) $
because
$ \rho _1^{-1}(A) \equiv _{\mathsf {W}}^Y \rho _2^{-1}(A) $
. Clearly,
$ \boldsymbol {\Gamma } \in \mathrm {Class}_{\operatorname {NSD}}(X) \iff \rho ^{-1}(\boldsymbol {\Gamma }) \in \mathrm {Class}_{\operatorname {NSD}}(Y) $
.
Notice that any boldface pointclass
$ \boldsymbol {\Gamma } \neq \{ \emptyset \} $
completely determines the space
$ X $
where it “lives,” as
$ X $
can be canonically recovered as the
$ \subseteq $
-largest set in
$ \boldsymbol {\Gamma } $
. This allows us to greatly simplify the notation: the symbol
$ \mathsf {W} $
is unnecessaryFootnote
5
because the order relation for pointclasses is simply inclusion; and as observed we can (almost) always drop the reference to the ambient space
$ X $
. So, for example, when
$ \boldsymbol {\Gamma } \neq \{ \emptyset \} $
is a Wadge class we can simply write
$ ||\boldsymbol {\Gamma }|| $
to denote the Wadge rank of
$ \boldsymbol {\Gamma } $
, i.e., the ordinal
$ ||A||_{\mathsf {W}}^X $
for some/any
$ A \subseteq X $
such that
$ \boldsymbol {\Gamma } = A {\downarrow }_X $
.
The chosen notation highlights the tight connection between ordinals, sets, and Wadge classes in a zero-dimensional Polish space
$ X $
: if
$ \boldsymbol {\Gamma } $
is a Wadge class generated by
$ A \subseteq X $
, then
$$\begin{align*}\boldsymbol{\Gamma} \in \mathrm{Class}_{\operatorname{NSD}}(X) \iff A \in \mathrm{Set}_{\operatorname{NSD}}(X) \iff ||\boldsymbol{\Gamma}|| = ||A||_{\mathsf{W}}^X \in \mathrm{Ord}_{\mathrm{NSD}}(X). \end{align*}$$
2.4 The relativization method
Building on previous work by Louveau and Saint-Raymond, the paper [Reference Carroy, Medini and Müller11] develops a substantial part of Wadge theory for arbitrary zero-dimensional Polish spaces, often in terms of Wadge classes. Here we recall the few facts that we need for the present work.
The relativization method, introduced in [Reference Louveau, Saint-Raymond, Kechris, Martin and Steel20], provides a way to “transfer” boldface pointclasses in
$ \mathcal {N} $
to the space
$ X $
(see [Reference Carroy, Medini and Müller11, Section 6]).
Definition 2.8. Let X be a zero-dimensional Polish space, and consider any boldface pointclass
. The relativization of
$ \boldsymbol {\Gamma } $
to
$ X $
is the boldface pointclass
$$\begin{align*}\boldsymbol{\Gamma}(X)=\{A \subseteq X \mid g^{-1}(A)\in\boldsymbol{\Gamma} \text{ for all continuous } g \colon \mathcal{N} \to X \}. \end{align*}$$
Notice that the dual (in
$ X $
) of
$ \boldsymbol {\Gamma }(X) $
is simply
, i.e., the relativization of the dual (in
$ \mathcal {N} $
) of
$ \boldsymbol {\Gamma } $
. Thus if
$ \boldsymbol {\Gamma }(X) \in \mathrm {Class}_{\operatorname {NSD}}(X) $
then
$ \boldsymbol {\Gamma } \in \mathrm {Class}_{\operatorname {NSD}}(\mathcal {N}) $
. In contrast, there are situations where
$ \boldsymbol {\Gamma } \in \mathrm {Class}_{\operatorname {NSD}}(\mathcal {N}) $
but
$ \boldsymbol {\Gamma }(X) \in \mathrm {Class}_{\operatorname {SD}}(X) $
, see the end of this subsection for more details.
The relativization method works particularly well with nonselfdual Wadge classes. The following result sums up the content of [Reference Carroy, Medini and Müller11, (Proof of) Lemma 6.5 and Theorem 7.2].
Theorem 2.9 (
$ \mathsf {AD} $
).
Let X be a zero-dimensional Polish space.
-
(1) For all
$\boldsymbol {\Lambda }\in \mathrm {Class}_{\operatorname {NSD}}(X)$
there is a unique boldface pointclass
such that
$\boldsymbol {\Lambda }=\boldsymbol {\Gamma }(X)$
, and moreover
$ \boldsymbol {\Gamma }\in \mathrm {Class}_{\operatorname {NSD}}(\mathcal {N}) $
. -
(2) If moreover X is uncountable, then
$\mathrm {Class}_{\operatorname {NSD}}(X)=\{\boldsymbol {\Gamma }(X)\mid \boldsymbol {\Gamma }\in \mathrm {Class}_{\operatorname {NSD}}(\mathcal {N})\}$
, and indeed
$ \langle \mathrm {Class}_{\operatorname {NSD}}(\mathcal {N}), {\subseteq } \rangle $
and
$ \langle \mathrm {Class}_{\operatorname {NSD}}(X), {\subseteq } \rangle $
are isomorphic via the map
$ \boldsymbol {\Gamma } \mapsto \boldsymbol {\Gamma }(X) $
.
Theorem 2.9(2) relies on the fact that if
$ \boldsymbol {\Gamma } \in \mathrm {Class}_{\operatorname {NSD}}(\mathcal {N}) $
then
$ \boldsymbol {\Gamma }(X) $
is still nonselfdual if
$ X $
is uncountable. In countable spaces this ceases to be true: since all sets in a countable Polish space X are
$\boldsymbol {\Delta }^0_2$
, all pointclasses in
$\mathrm {Class}_{\operatorname {NSD}}(\mathcal {N})$
containing
$\boldsymbol {\Delta }^0_2$
relativize to
, which is selfdual. We will provide the correct analogue of Theorem 2.9(2) for countable Polish spaces in Corollary 3.19.
2.5 Cantor–Bendixson derivatives and rank
Recall from [Reference Kechris17, Section 6.C] the classical notion of Cantor–Bendixson derivative, defined by
$$\begin{align*}D_{\operatorname{CB}}(Y) =\{x \in Y \mid x \text{ is not isolated in }Y\}, \end{align*}$$
and its iterates
$ D^{\alpha }_{\operatorname {CB}}(X) $
, recursively defined by setting
$ D^{0}_{\operatorname {CB}}(X) = X $
,
$ D^{\alpha +1}_{\operatorname {CB}}(X) = D_{\operatorname {CB}}(D^{\alpha }_{\operatorname {CB}}(X)) $
, and
$ D^{\lambda }_{\operatorname {CB}}(X) = \bigcap _{\alpha < \lambda } D^{\alpha }_{\operatorname {CB}}(X) $
for
$\lambda $
limit. When
$ X $
is Polish, there is
$ \alpha < \omega _1 $
such that
$ D^{\alpha }_{\operatorname {CB}}(X) = D^{\alpha +1}_{\operatorname {CB}}(X) $
: the smallest such
$ \alpha $
is the Cantor–Bendixson rank (CB-rank for short) of
$ X $
and is denoted by
$ ||X||_{\operatorname {CB}} $
, while
$\operatorname {ker}_{\operatorname {CB}}(X) = D^{||X||_{\operatorname {CB}}}_{\operatorname {CB}}(X)$
is called perfect kernel of
$ X $
and can equivalently be defined as the largest perfectFootnote
6
subspace of
$ X $
. Moreover, for
$x \in X \setminus \operatorname {ker}_{\operatorname {CB}}(X)$
we let
$||x,X||_{\operatorname {CB}}$
be the unique ordinal
$\alpha <||X||_{\operatorname {CB}}$
satisfying
$x \in D^{\alpha }_{\operatorname {CB}}(X) \setminus D^{\alpha +1}_{\operatorname {CB}}(X)$
.
If
$ X $
is countable, then
$ \operatorname {ker}_{\operatorname {CB}}(X) = \emptyset $
. In this case, we define the Cantor–Bendixson degree
$ \mathrm {Deg}_{\operatorname {CB}}(X) $
of X (CB-degree for short) as the cardinality of
$D^{\alpha }_{\operatorname {CB}}(X)$
if
$||X||_{\operatorname {CB}}=\alpha +1$
is a successor ordinal, and
$\mathrm {Deg}_{\operatorname {CB}}(X)=\omega $
otherwise. The Cantor–Bendixson type of X (CB-type for short) is the pair
$ \operatorname {\mathrm {tp}}(X) = ( ||X||_{\operatorname {CB}} , \mathrm {Deg}_{\operatorname {CB}}(X)) $
. For compact countable Polish spaces,
$ \operatorname {\mathrm {tp}}(X) $
is a complete invariant for homeomorphism.
We say that a countable Polish space
$ X $
is simple if it has successor CB-rank
$||X||_{\operatorname {CB}} = \alpha +1$
and
$ \mathrm {Deg}_{\operatorname {CB}}(X) = 1 $
, that is, if
$D^{\alpha }_{\operatorname {CB}}(X)$
is a singleton. The following is a special case of [Reference Carroy9, Lemma 2.4], and shows in particular that if
$ X $
is a countable Polish space with limit CB-rank, then
$ X $
is not compact.
Lemma 2.10. Every countable Polish space X of CB-type
$(\alpha ,\beta ) \in \omega _1\times (\omega +1)$
admits a partition in nonempty clopen subsets
$(C_n)_{n \in \beta }$
such that each
$C_n$
is simple,
$||C_n||_{\operatorname {CB}}=\alpha $
if
$\alpha $
is successor, and
$||C_n||_{\operatorname {CB}}<\alpha $
with
$ \sup _{n \in \omega } ||C_n||_{\operatorname {CB}} = \alpha $
if
$\alpha $
is limit.
The following is instead a direct application of [Reference Carroy9, Corollary 2.3].
Lemma 2.11. Let X be a zero-dimensional Polish space and
$U \subseteq X$
be an open set such that
$ U \cap \operatorname {ker}_{\operatorname {CB}}(X) = \emptyset $
. Then
$||U||_{\operatorname {CB}}= \min \{ \alpha \leq ||X||_{\operatorname {CB}}\mid U \cap D^{\alpha }_{\operatorname {CB}}(X)= \emptyset \}$
.
2.6 The pointed gluing operation
We adopt a standard notation for sequences: for example,
$ \operatorname {lh}(s) $
denotes the length of
$ s $
,
$s {^{\smallfrown }} t$
stands for the concatenation of s and t, while for
$ i \in \omega $
and
$ n \in \omega $
we denote by
$ (i)^n $
(respectively,
$ (i)^{\infty } $
) the constant sequence of length
$ n $
(respectively, the
$ \omega $
-sequence) with value
$ i $
. If X is a topological space, we say that a sequence
$(A_n)_{n \in \omega }$
of subsets of
$ X $
converges to a point
$ x \in X $
, in symbols
$A_n \to x$
, if any open neighborhood of x contains all but finitely many
$A_n$
’s.
The following construction, taken from [Reference Carroy9, Section 3], is implicitly used in the area since Wadge’s thesis [Reference Wadge33] and has been considered, sometimes with slight variations, by several authors (see, e.g., [Reference Selivanov27, Section 5]). Given a sequence
$(A_n)_{n \in \omega }$
with
$A_n \subseteq \mathcal {N}$
, its pointed gluing is the set
$$ \begin{align*} \operatorname{ptgl}((A_n)_{n \in \omega}) = (0)^{\infty} \cup \bigcup_{n \in \omega}\{{(0)^{n}}{^{\smallfrown}}(1){^{\smallfrown}} x \mid x \in A_n\} \end{align*} $$
Intuitively,
$ \operatorname {ptgl}((A_n)_{n \in \omega }) $
is thus obtained by taking copies of the
$ A_n $
’s which converge to
$ (0)^{\infty } $
, together with such limit. The functional counterpart of the pointed gluing operation is defined by considering a sequence
$ (f_n)_{n \in \omega } $
of functions
$ f_n \colon A_n \to B_n $
, and letting
$$\begin{align*}\operatorname{ptgl}((f_n)_{n \in \omega}) \colon \operatorname{ptgl}((A_n)_{n \in \omega}) \to \operatorname{ptgl}((B_n)_{n \in \omega}) \end{align*}$$
be defined as the map sending
$ (0)^{\infty } $
to itself and points of the form
$ (0)^n{^{\smallfrown }}(1){^{\smallfrown }} x $
with
$ x \in A_n $
to
$ (0)^n{^{\smallfrown }}(1){^{\smallfrown }} f_n(x) $
. When
$A_n = A $
or
$ f_n = f $
for all
$n \in \omega $
, to simplify the notation we write
$\operatorname {ptgl}(A)$
instead of
$\operatorname {ptgl}((A_n)_{n \in \omega })$
or
$ \operatorname {ptgl}(f) $
instead of
$ \operatorname {ptgl}((f_n)_{n \in \omega }) $
, respectively.
It is easy to verify that if
$A_n$
is closed (respectively, compact), then so is
$\operatorname {ptgl}((A_n)_{n \in \omega })$
, and that
$\operatorname {ptgl}((f_n)_{n \in \omega })$
is continuous (respectively, injective, an embedding, or a homeomorphism) if so are the
$ f_n $
’s. Moreover, if the sequence of ordinals
$(||A_n||_{\operatorname {CB}})_{n\in \omega }$
is non-decreasing, then an easy computation shows that
$||\operatorname {ptgl}((A_n)_{n \in \omega })||_{\operatorname {CB}}=(\sup _{n \in \omega }||A_n||_{\operatorname {CB}})+1$
(see, e.g., [Reference Carroy9, Proposition 3.1]).
2.7 Difference hierarchy
We recall the definition of difference hierarchy (see, e.g., [Reference Kechris17, Section 22.E]). Let X be a Polish space,
$\alpha \geq 1$
be a countable ordinal, and
$(A_{\gamma })_{\gamma <\alpha }$
be a sequence of subsets of
$ X $
. Then we define
$$\begin{align*}D_{\alpha}((A_{\gamma})_{\gamma < \alpha}) = \bigcup \bigg \{A_{\gamma} \setminus \bigcup_{\gamma' < \gamma} A_{\gamma'} \mid \gamma < \alpha \text{ and } \gamma \text{ has parity opposite to }\alpha\bigg\}. \end{align*}$$
Moreover, when the boldface pointclass
is different from
$ \{ \emptyset \} $
we set
$ D_{\alpha }(\boldsymbol {\Gamma }) = \{D_{\alpha }((A_{\gamma })_{\gamma < \alpha }) \mid A_{\gamma }\in \boldsymbol {\Gamma }\}$
. By convention, we set
$D_{0}(\boldsymbol {\Gamma }) = \{\emptyset \}$
. Notice also that
$ D_{1}(\boldsymbol {\Gamma }) = \boldsymbol {\Gamma } $
.
We will always work with
$ \boldsymbol {\Gamma } = \boldsymbol {\Sigma }^0_1(X) $
. One can easily verify that
$ D_{\alpha }(\boldsymbol {\Sigma }^0_1(X)) $
is a boldface pointclass (in
$ X $
), and that
$ D_{\alpha }(\boldsymbol {\Sigma }^0_1(X)) = D_{\alpha }(\boldsymbol {\Sigma }^0_1)(X) $
. Thus we can unambiguously denote by
the dual of
$ D_{\alpha }(\boldsymbol {\Sigma }^0_1(X)) $
.
Wadge himself proved in [Reference Wadge33] that the first
$ \omega _1 $
-many nonselfdual levels of
are occupied by the difference classes
$ D_{\alpha }(\boldsymbol {\Sigma }^0_1) $
,
$ \alpha < \omega _1 $
, and their duals. It follows that
$ ||D_{\alpha }(\boldsymbol {\Sigma }^0_1)||_{\mathcal {N}} = 2 \alpha +1 $
, and that using the well-known successor and countable sum operations from [Reference Wadge33, Chapter 3] one can define complete sets (up to Wadge equivalence) for such classes. More precisely, we recursively define a sequence of Wadge degrees
$ (\mathbf {D}_\alpha )_{\alpha < \omega _1} $
by setting
$$ \begin{align} \mathbf{D}_\alpha = \begin{cases} [\emptyset]_{\mathsf{W}}^{\mathcal{N}}, & \text{if }\alpha = 0, \\ [ \{ (0)^n {^{\smallfrown}} (m+1) {^{\smallfrown}} x \mid n,m \in \omega \wedge x \in \neg B_\beta \}]_{\mathsf{W}}^{\mathcal{N}}, & \text{if }\alpha = \beta+1, \\ [ \{ (0)^n {^{\smallfrown}} (m+1) {^{\smallfrown}} x \mid n,m \in \omega \wedge x \in \neg B_{\alpha_n} \}]_{\mathsf{W}}^{\mathcal{N}}, & \text{if } \alpha \text{ is limit,} \end{cases} \end{align} $$
where
$ B_\beta $
is any set in
$ \mathbf {D}_\beta $
,
$ (\alpha _n)_{n \in \omega }$
is some/any increasing sequence cofinal in
$\alpha $
, and
$ B_{\alpha _n} $
is an arbitrary set in
$ \mathbf {D}_{\alpha _n} $
. An easy induction shows that the sequence
$ (\mathbf {D}_\alpha )_{\alpha < \omega _1} $
is well-defined, i.e., that up to
$ \equiv _{\mathsf {W}}^{\mathcal {N}} $
it is independent of the choice of
$ B_\beta $
,
$(\alpha _n)_{n \in \omega } $
, and
$ B_{\alpha _n} $
. Moreover, one can prove by induction again that if
$ B_\alpha \in \mathbf {D}_\alpha $
then
$ D_{\alpha }(\boldsymbol {\Sigma }^0_1) = (B_\alpha ) {\downarrow }_{\mathcal {N}} $
.
3 Our toolbox
3.1 More on the retraction method
The retraction method immediately settles our problem for zero-dimensional Polish spaces which are not
$ \sigma $
-compact, i.e., cannot be written as a countable union of compact spaces, and for those which are uncountable and compact. This shows that we could in principle concentrate on
$ \sigma $
-compact non-compact spaces, although our analysis will be more general and work for all zero-dimensional Polish spaces, without those limitations.
The next two results do not require any extra assumption beyond our base theory
$ \mathsf {ZF} + \mathsf {DC}(\mathbb {R}) $
.
Lemma 3.1. Suppose that
$ X,Y $
are zero-dimensional Polish spaces, each of which is homeomorphic to closed subset of the other one. Then
and
are isomorphic.
Proof Without loss of generality,
$ X $
is a closed subset of
$ Y $
and
$ Y $
is a closed subspace of some
$ X ' $
which is homeomorphic to
$ X $
via some
$ h \colon X \to X' $
. Let
$ \rho _1 \colon Y \to X $
and
$ \rho _2 \colon X' \to Y $
be retractions. We already observed that
$ A \mapsto \rho _1^{-1} (A) $
provides an embedding of
into
, and hence it canonically induces an embedding of
into
: we claim that the latter is also surjective (hence an isomorphism), i.e., that for every
$ B \subseteq Y $
there is
$ A \subseteq X $
such that
$ B \equiv _{\mathsf {W}}^Y \rho _1^{-1}(A) $
. To see this, it is enough to notice that since
$ \rho _1 $
and
$ \rho _2 $
are retractions and
$ h $
is a homeomorphism, then
$$\begin{align*}B \equiv_{\mathsf{W}}^{Y,X'} \rho_2^{-1}(B) \equiv_{\mathsf{W}}^{X',X} h^{-1}(\rho_2^{-1}(B)) \equiv_{\mathsf{W}}^{X,Y} \rho_1^{-1}(h^{-1}(\rho_2^{-1}(B))). \end{align*}$$
Thus setting
$ A = h^{-1}(\rho _2^{-1}(B)) $
we are done.
Combining the above lemma with the universality properties of
$ \mathcal {N} $
and
$ \mathcal {C} $
[Reference Kechris17, Theorem 7.8], the Cantor–Bendixson theorem [Reference Kechris17, Corollary 6.5] and the Hurewicz theorem [Reference Kechris17, Theorem 7.10] we obtain the following.
Proposition 3.2. Let
$ X $
be a zero-dimensional Polish space.
-
(1) If
$ X $
is not
$\sigma $
-compact,Footnote
7
then
is isomorphic to
. -
(2) If
$ X $
is compact and uncountable, then
is isomorphic to
.
Remark 3.3. Since
$\sigma $
-compactness already appeared in [Reference Motto Ros, Schlicht and Selivanov24, Theorem 5.12(2)] as a dividing line for Wadge-like hierarchies on zero-dimensional Polish space, one may be tempted to conjecture that Proposition 3.2(1) can be turned into a characterization. We will however show that this is not the case: by Main Theorem 1, there are
$\sigma $
-compact spaces whose Wadge hierarchy is isomorphic to
(take any
$\sigma $
-compact space with non-compact perfect kernel, e.g.,
$ \omega \times \mathcal {C} $
), and there are non-compact spaces whose Wadge hierarchy is isomorphic to
(take any non-compact space with compact kernel and finite CB-rank, e.g.,
$ \omega \oplus \mathcal {C} $
where
$ \oplus $
denotes the disjoint sum and
$ \omega $
is discrete).
3.2 More on the relativization method
The next lemma shows that for any retraction
$ \rho \colon \mathcal {N} \to X $
, the map
$ \boldsymbol {\Gamma } \mapsto \rho ^{-1}(\boldsymbol {\Gamma }) $
from
$ \mathrm {Class}_{\mathsf {W}}(X) $
to
$ \mathrm {Class}_{\mathsf {W}}(\mathcal {N}) $
is a right inverse of (the restriction to
$ \mathrm {Class}_{\mathsf {W}}(\mathcal {N}) $
of) the relativization map
$ \boldsymbol {\Gamma } \mapsto \boldsymbol {\Gamma }(X) $
. The proof is the same as the first part of the proof of [Reference Carroy, Medini and Müller11, Lemma 6.5] and does not require extra assumptions beyond
$ \mathsf {ZF}+ \mathsf {DC}(\mathbb {R}) $
.
Lemma 3.4. Let
$ X \subseteq \mathcal {N} $
be closed, and
$ \rho \colon \mathcal {N} \to X $
be a retraction. Then for every
$ \boldsymbol {\Lambda } \in \mathrm {Class}_{\mathsf {W}}(X), $
$$\begin{align*}(\rho^{-1}(\boldsymbol{\Lambda}))(X) = \boldsymbol{\Lambda}. \end{align*}$$
Proof Let
$ A \subseteq X $
be complete for
$ \boldsymbol {\Lambda } $
. If
$ B \in (\rho ^{-1}(\boldsymbol {\Lambda }))(X) $
then
$ \rho ^{-1}(B) \in \rho ^{-1}(\boldsymbol {\Lambda }) $
because
$ \rho $
is continuous, thus
$ \rho ^{-1}(B) \le _{\mathsf {W}}^{\mathcal {N}} \rho ^{-1}(A) $
by definition of
$ \rho ^{-1}(\boldsymbol {\Lambda }) $
, hence
$ B \le _{\mathsf {W}}^X A $
, and finally
$ B \in \boldsymbol {\Lambda } $
by choice of
$ A $
.
Conversely, pick any
$ B \in \boldsymbol {\Lambda } $
, let
$ f \colon X \to X $
witness
$ B \le _{\mathsf {W}}^X A $
, and let
$ g \colon \mathcal {N} \to X $
be any continuous function. Then
$ f \circ g $
witnesses
$ g^{-1}(B) \le _{\mathsf {W}}^{\mathcal {N},X} A $
, and since
$ A \equiv _{\mathsf {W}} ^{X,\mathcal {N}} \rho ^{-1}(A) $
we get
$ g^{-1}(B) \le _{\mathsf {W}}^{\mathcal {N}} \rho ^{-1}(A) $
, i.e.,
$ g^{-1}(B) \in \rho ^{-1}(\boldsymbol {\Lambda }) $
. Since
$ g $
was arbitrary, this shows that
$ B \in (\rho ^{-1}(\boldsymbol {\Lambda }))(X) $
, as desired.
Corollary 3.5 (
$ \mathsf {AD} $
).
Let
$ X \subseteq Y $
be zero-dimensional Polish spaces with
$ X $
closed in
$ Y $
, and let
be a boldface pointclass. If
$ \boldsymbol {\Gamma }(X) \in \mathrm {Class}_{\operatorname {NSD}}(X) $
, then
$ \boldsymbol {\Gamma }(Y) \in \mathrm {Class}_{\operatorname {NSD}}(Y) $
. Moreover, if
$ A \subseteq X $
is complete for
$ \boldsymbol {\Gamma }(X) $
and
$ \rho \colon Y \to X $
is any retraction, then
$ \rho ^{-1}(A) $
is complete for
$ \boldsymbol {\Gamma }(Y) $
.
Proof Without loss of generality, we may assume that
$ Y $
(and hence also
$ X $
) is a closed subset of
$ \mathcal {N} $
. Let
$ A $
and
$ \rho $
be as in the statement, and fix a retraction
$ \rho ' \colon \mathcal {N} \to Y $
, so that
$ \rho \circ \rho ' $
is a retraction of
$ \mathcal {N} $
onto
$ X $
. Then
$ (\rho \circ \rho ')^{-1}(\boldsymbol {\Gamma }(X)) = \boldsymbol {\Gamma } $
by Lemma 3.4 applied to
$ \boldsymbol {\Lambda } = \boldsymbol {\Gamma }(X) $
and Theorem 2.9(1). Using this and applying once again Lemma 3.4 with
$ \boldsymbol {\Lambda } = \rho ^{-1}(\boldsymbol {\Gamma }(X)) \in \mathrm {Class}_{\mathsf {W}}(Y) $
, it follows that
$$\begin{align*}\rho^{-1}(\boldsymbol{\Gamma}(X)) = ((\rho')^{-1}(\rho^{-1}(\boldsymbol{\Gamma}(X))))(Y) = ((\rho \circ \rho')^{-1}(\boldsymbol{\Gamma}(X))) (Y) = \boldsymbol{\Gamma}(Y) , \end{align*}$$
which proves at once that
$ \boldsymbol {\Gamma }(Y) $
is nonselfdual (because so is
$ \rho ^{-1}(\boldsymbol {\Gamma }(X)) $
) and that
$ \rho ^{-1}(A) $
is complete for
$ \boldsymbol {\Gamma }(Y) $
.
The relativization method and, in particular, Corollary 3.5, gives a general version of the classical computation in
$\mathcal {N}$
of the supremum of at most countably many nonselfdual classes, that we use several times in this paper.
Lemma 3.6 (
$ \mathsf {AD} $
).
Let X be a zero-dimensional Polish space,
$(X_i)_{i\in I}$
be a clopen partition of X for some
$I\subseteq \omega $
, and
$(\boldsymbol {\Gamma }_i)_{i\in I}$
be boldface pointclasses in
$\mathcal {N}$
. Suppose that
$\boldsymbol {\Gamma }_i(X_i) \in \mathrm {Class}_{\operatorname {NSD}}(X_i)$
for all
$ i \in I $
, and let
$A_i\subseteq X_i$
be complete for
$\boldsymbol {\Gamma }_i(X_i)$
. Then the pointclass
$\boldsymbol {\Gamma }=A {\downarrow }_X \in \mathrm {Class}_{\mathsf {W}}(X)$
generated by
$A=\bigcup _{i \in I} A_i$
is the smallest Wadge class such that
$\boldsymbol {\Gamma } \supseteq \boldsymbol {\Gamma }_i(X)$
for all
$ i \in I $
.
Moreover, if for every
$ i \in I $
there is
$ j \in I $
such that
, then
$ \boldsymbol {\Gamma } \in \mathrm {Class}_{\operatorname {SD}}(X) $
.
The class
$ \boldsymbol {\Gamma }=A {\downarrow }_X $
from the lemma is called the supremum of the
$\boldsymbol {\Gamma }_i$
’s in X.
Proof For each
$ i \in I $
pick a retraction
$ \rho _i \colon X \to X_i $
, so that
$ \boldsymbol {\Gamma }_i(X) = \rho _i^{-1}(A_i) {\downarrow }_X $
by Corollary 3.5. Since the identity function witnesses
$ A_i \le _{\mathsf {W}}^{X_i,X} A $
and
$\rho _i^{-1}(A_i) \equiv _{\mathsf {W}}^{X,X_i} A_i $
, we get
$ \boldsymbol {\Gamma }_i(X) \subseteq A {\downarrow }_X = \boldsymbol {\Gamma } $
. Moreover, if
$ B \subseteq X $
is such that
$ B {\downarrow }_X \supseteq \boldsymbol {\Gamma }_i(X) $
, then
$ \rho _i^{-1}(A_i) \le _{\mathsf {W}}^X B $
, and hence
$ A_i \le _{\mathsf {W}}^{X_i,X} B $
via some
$ f_i \colon X_i \to X $
. It follows that
$ \bigcup _{i \in I } f_i $
witnesses
$ A \le _{\mathsf {W}}^X B $
, and thus
$ \boldsymbol {\Gamma } \subseteq B {\downarrow }_X $
.
For the second part, notice that we can assume without loss of generality that
$ X $
is closed in
$ \mathcal {N} $
, so that the
$ X_i $
’s are closed in
$ \mathcal {N} $
as well. Let
$ r_i \colon \mathcal {N} \to X_i $
be retractions. If
$ i,j \in I $
are as in the statement, then using Corollary 3.5 again we get
$$\begin{align*}X_i \setminus A_i \equiv_{\mathsf{W}}^{X_i,\mathcal{N}} {\mathcal{N} \setminus r_i^{-1}(A_i)} \le_{\mathsf{W}}^{\mathcal{N}} r_{j}^{-1}(A_{j}) \equiv_{\mathsf{W}}^{\mathcal{N},X_{j}} A_{j}, \end{align*}$$
where the middle reduction exists because
. If
$ f_i \colon X_i \to X_{j} $
witnesses
$ X_i \setminus A_i \le _{\mathsf {W}}^{X_i,X_{j}} A_{j} $
, then
$ \bigcup _{i \in I} f_i $
witnesses
$ \neg A \le _{\mathsf {W}}^X A $
, as desired.
The typical situations in which we will apply the “moreover” part of Lemma 3.6 are when
$I = 2$
and
, or
$I=\omega $
and
$\boldsymbol {\Gamma }_i \subsetneq \boldsymbol {\Gamma }_{i+1}$
(in the second case use Wadge’s Lemma to obtain
). Notice also that if X is uncountable, then by Lemma 2.9(2) the hypothesis
$\boldsymbol {\Gamma }_i(X) \in \mathrm {Class}_{\operatorname {NSD}}(X_i)$
follows from the more manageable
$ \boldsymbol {\Gamma }_i \in \mathrm {Class}_{\operatorname {NSD}}(\mathcal {N}) $
.
3.3 Compact rank
An important dividing line in our analysis is whether the given Polish space
$X $
has a compact perfect kernel or not. If
$ X $
has a compact perfect kernel, then its iterated Cantor–Bendixson derivatives are eventually compact. We use this idea to define a new ordinal invariant, called compact rank.
Definition 3.7. Let
$ X $
be a Polish space such that
$ \operatorname {ker}_{\operatorname {CB}}(X) $
is compact. Then the compact rank of
$ X $
is
$$\begin{align*}||X||_{\operatorname{Comp}} = \min \{\alpha \leq ||X||_{\operatorname{CB}} \mid D^{\alpha}_{\operatorname{CB}}(X) \text{ is compact}\}. \end{align*}$$
The assumption on
$ \operatorname {ker}_{\operatorname {CB}}(X) $
ensures that
$ ||X||_{\operatorname {Comp}} $
is well-defined. Moreover,
$ ||X||_{\operatorname {Comp}} = 0 $
if and only if the whole
$ X $
is compact.
There is a natural characterization of when a space with compact perfect kernel is not compact itself which will be useful later on. It is based on the following well-known exercise.Footnote 8
Fact 3.8. A zero-dimensional second-countable space is not compact if and only if it admits an infinite (countable) clopen partition.
Lemma 3.9. Suppose that X is a zero-dimensional Polish space with
$\operatorname {ker}_{\operatorname {CB}}(X)$
compact. Then X is not compact if and only if there exists an infinite discrete clopen set
$U \subseteq X$
.
Proof If U is infinite discrete then it is not compact, and if it is closed in
$ X $
then
$ X $
cannot be compact either.
Conversely, suppose that X is not compact. By Fact 3.8, there exists an infinite clopen partition
$(C_n)_{n \in \omega }$
of X. Since
$\operatorname {ker}_{\operatorname {CB}}(Z)$
is compact, there exists
$m \in \omega $
such that
$\operatorname {ker}_{\operatorname {CB}}(Z)\subseteq \bigcup _{n\leq m}C_n$
. For each
$n>m$
there is at least one isolated point
$ x_n \in C_n $
because
$ C_n \cap \operatorname {ker}_{\operatorname {CB}}(X) = \emptyset $
: then
$U = \{x_n \mid n> m\}$
is as required.
It might be interesting to notice that the compact rank can be obtained through a corresponding derivative
$ D_{\operatorname {Comp}}(F) $
defined by
$$ \begin{align*} D_{\operatorname{Comp}}(F) = F \setminus \bigcup\{U \mid U\text{ is infinite discrete clopen in }F\}. \end{align*} $$
(With this terminology, Lemma 3.9 can be reformulated as: If
$ \operatorname {ker}_{\operatorname {CB}}(X) $
is compact, then X is compact if and only if
$D_{\operatorname {Comp}}(X)=X$
.) One can then prove that if X is a non-compact zero-dimensional Polish space with
$\operatorname {ker}_{\operatorname {CB}}(X)$
compact, then
$D_{\operatorname {CB}}(X) = D_{\operatorname {Comp}}(X)$
, and thus
$ ||X||_{\operatorname {Comp}} $
is the rank associated with the derivative
$ D_{\operatorname {Comp}} $
, i.e.,
$ ||X||_{\operatorname {Comp}} = \min \{\alpha < \omega _1 \mid D^{\alpha }_{\operatorname {Comp}}(X) = D^{\alpha +1}_{\operatorname {Comp}}(X) \}. $
We will also need the following technical lemma.
Lemma 3.10. Suppose that X is a non-compact zero-dimensional Polish space with
$\operatorname {ker}_{\operatorname {CB}}(X)$
compact. Then for all
$\alpha < ||X||_{\operatorname {Comp}}$
, there exists an infinite clopen partition
$(C_n)_{n \in \omega }$
of X such that
$C_0 \supseteq D^{\alpha +1}_{\operatorname {CB}}(X)$
and
$||C_{n+1}||_{\operatorname {CB}}=\alpha +1$
for all
$n\in \omega $
.
Proof Since
$\alpha < ||X||_{\operatorname {Comp}} \leq ||X||_{\operatorname {CB}}$
we have
$D^{\alpha +1}_{\operatorname {CB}}(X) \subsetneq D^{\alpha }_{\operatorname {CB}}(X)$
and the latter is not compact. By Lemma 3.9 there is an infinite discrete clopen (relatively to
$ D^{\alpha }_{\operatorname {CB}}(X) $
) set
$U = \{ x_i \mid i \in \omega \} \subseteq D^{\alpha }_{\operatorname {CB}}(X)$
, so that
$ U \cap D^{\alpha +1}_{\operatorname {CB}}(X) = \emptyset $
. Let
$ (U_n)_{n \in \omega } $
be open sets of
$ X $
such that
$ U_0 \cap D^{\alpha }_{\operatorname {CB}}(X) = D^{\alpha }_{\operatorname {CB}}(X) \setminus U $
and
$ U_{i+1} \cap D^{\alpha }_{\operatorname {CB}}(X) = \{ x_i \} $
. Without loss of generality, we may assume that
$ \bigcup _{n \in \omega } U_n = X $
(otherwise we replace
$ U_0 $
with
$ U_0 \cup (X \setminus D^{\alpha }_{\operatorname {CB}}(X) $
). Using the generalized reduction property [Reference Kechris17, Theorem 22.16] we obtain a clopen partition
$ (C_n)_{n \in \omega } $
of
$ X $
such that
$ C_n \subseteq U_n $
, so that in particular
$ C_n \cap D^{\alpha }_{\operatorname {CB}}(X) = U_n \cap D^{\alpha }_{\operatorname {CB}}(X) $
because
$ (U_n \cap D^{\alpha }_{\operatorname {CB}}(X))_{n \in \omega } $
is a clopen partition of
$ D^{\alpha }_{\operatorname {CB}}(X) $
. Since
$ U \cap D^{\alpha +1}_{\operatorname {CB}}(X) = \emptyset $
, by Lemma 2.11 the clopen partition
$(C_n)_{n \in \omega }$
of
$ X $
is as desired.
3.4 Minimal countable spaces
We define (up to homeomorphism) a canonical sequence
$ (K_{\alpha +1})_{\alpha < \omega _1} $
of countable compact metrizable spaces such that each
$ K_{\alpha +1} $
has CB-type
$ (\alpha +1,1) $
(thus it is simple) and embeds into every Polish space of CB-rank at least
$ \alpha +1 $
.
Definition 3.11. Set
$K_0=\emptyset $
and inductively define the space
$K_{\alpha +1}$
for all
$\alpha < \omega _1$
as follows:
-
•
$K_{\alpha +1} = \operatorname {ptgl}(K_{\alpha })$
, when
$\alpha = 0$
or
$\alpha $
is a successor ordinal. -
•
$K_{\alpha +1} = \operatorname {ptgl}((K_{\alpha _n+1})_{n \in \omega })$
for some
$(\alpha _n)_{n \in \omega }$
cofinal in
$\alpha $
, when
$\alpha $
is limit.
Note that
$K_1=\{(0)^{\infty }\}$
. By induction on
$\alpha <\omega _1$
, one easily sees that
$ K_{\alpha +1} $
is compact and
$\operatorname {\mathrm {tp}}(K_{\alpha +1})=(\alpha +1,1)$
. As a consequence, since the CB-type is a complete invariant for compact countable Polish spaces the definition of
$K_{\alpha +1}$
when
$\alpha $
is limit does not depend on the choice of the cofinal sequence
$(\alpha _n)_{n\in \omega }$
, so the sequence is well-defined up to homeomorphism.Footnote
9
Proposition 3.12. For every ordinal
$\alpha < \omega _1$
and every Polish space X with
$||X||_{\operatorname {CB}} \geq \alpha + 1$
there exists an embedding
$i \colon K_{\alpha + 1} \to X \setminus \operatorname {ker}_{\operatorname {CB}}(X)$
with compact (hence closed) range.
Proof Since
$ K_{\alpha +1} $
is compact, it is enough to prove by induction on
$\alpha $
that there is a continuous injection
$i \colon K_{\alpha + 1} \to X \setminus \operatorname {ker}_{\operatorname {CB}}(X)$
.
If
$ \alpha = 0 $
then
$ K_1 = \{ (0)^{\infty } \} $
and
$ X $
has at least one isolated point
$ x $
: setting
$ i((0)^{\infty }) = x $
we are done.
Assume now that
$ \alpha> 0 $
. Since
$ ||X||_{\operatorname {CB}} \geq \alpha +1 $
, then
$ D^{\alpha }_{\operatorname {CB}}(X) \setminus D^{\alpha +1}_{\operatorname {CB}}(X) \neq \emptyset $
. Pick any
$ x \in D^{\alpha }_{\operatorname {CB}}(X) \setminus D^{\alpha +1}_{\operatorname {CB}}(X) $
and fix any
$ \subseteq $
-decreasing clopen basis
$(V_n)_{n\in \omega }$
of neighborhoods of x with
$ V_0 \cap D^{\alpha }_{\operatorname {CB}}(X) = \{ x \} $
(hence also
$ V_0 \cap D^{\alpha +1}_{\operatorname {CB}}(X) = \emptyset $
). Set
$ U_n = V_n \setminus V_{n+1} $
, so that each
$ U_n $
is a countable clopen set,
$ U_n \to x $
,
$ ||U_n||_{\operatorname {CB}} \leq \alpha $
because
$ U_n \cap D^{\alpha }_{\operatorname {CB}}(X) = \emptyset $
, and
$ \limsup _{n \in \omega } ||U_n||_{\operatorname {CB}} = \alpha $
Footnote
10
because every clopen
$ x \in V \subseteq V_0 $
is such that
$ ||V||_{\operatorname {CB}} = \alpha +1 $
. (Here we repeatedly use Lemma 2.11.) Let
$ (\alpha _n)_{n \in \omega } $
be the constant sequence with value
$ \alpha $
if the latter is a successor ordinal, and a sequence of successor ordinals cofinal in
$\alpha $
otherwise. By
$ \limsup _{n \in \omega } ||U_n||_{\operatorname {CB}} = \alpha $
, there is an injective
$ f \colon \omega \to \omega $
such that
$ ||U_{f(n)}||_{\operatorname {CB}} \geq \alpha _n $
for all
$ n \in \omega $
, so that by inductive hypothesis there is a continuous injection
$ i_n \colon K_{\alpha _n} \to U_{f(n)} $
. Then the map
$ i \colon K_{\alpha +1} \to V_0 \subseteq X \setminus \operatorname {ker}_{\operatorname {CB}}(X) $
defined by
$$\begin{align*}i(y) = \begin{cases} x, & \text{if } y = (0)^{\infty}, \\ i_n(z), & \text{if } y = (0)^n {}^{\smallfrown{}} (1) {}^{\smallfrown{}} z \text{ with } z \in K_{\alpha_n}, \end{cases} \end{align*}$$
is the desired continuous injection.
3.5 More on the difference hierarchy
Section 2.7 and Theorem 2.9 yields a description of the first
$ \omega _1 $
-many nonselfdual Wadge classes in
when
$ X $
is uncountable. (Here we are not requiring
$ \mathsf {AD} $
because Borel determinacy suffices.)
Lemma 3.13. Let
$ X $
be an uncountable zero-dimensional Polish space and
$ \alpha < \omega _1 $
. Then the
$ \alpha $
th nonselfdual ordinal in
$ X $
is precisely the Wadge rank of
$ D_{\alpha }(\boldsymbol {\Sigma }^0_1(X)) $
.
Moving to countable Polish spaces
$ X $
, we instead observe that the difference hierarchy trivializes at least from
$ ||X||_{\operatorname {CB}} $
on. Recall that we only consider nonempty spaces
$ X $
.
Proposition 3.14. Let X be a countable Polish space. If
$||X||_{\operatorname {CB}} \le \alpha $
then
, and thus
. Moreover, if X is simple and
$||X||_{\operatorname {CB}}=\beta +1$
, then
.
Proof Since
$ D_{\gamma }(\boldsymbol {\Sigma }^0_1(X)) \subseteq D_{\gamma '}(\boldsymbol {\Sigma }^0_1(X)) $
if
$ \gamma \leq \gamma ' $
, it is enough to consider the case
$ ||X||_{\operatorname {CB}} = \alpha $
and prove by induction that
. Note that
$ 1 \leq \alpha < \omega _1 $
and
$D^{\alpha }_{\operatorname {CB}}(X)=\emptyset $
because
$ X \neq \emptyset $
is countable.
If
$\alpha = 1$
then X is discrete, so
. If moreover X is simple, then
$ X $
is a singleton and
.
Suppose now that
$\alpha = \beta + 1$
for some
$1 \leq \beta < \omega _1$
, and consider an arbitrary
$A\subseteq X$
. Let
$Y =X \setminus D^{\beta }_{\operatorname {CB}}(X)$
and
$A'=Y \setminus A$
. Since
$||Y||_{\operatorname {CB}} = \beta $
by Lemma 2.11, we get
$A' \in D_{\beta }(\boldsymbol {\Sigma }^0_1(Y))\subseteq D_{\beta }(\boldsymbol {\Sigma }^0_1(X))$
by inductive hypothesis and the fact that Y is open in X, so we can write
$ A' = D_{\beta }((U_\gamma )_{\gamma < \beta })$
with
$ U_\gamma \in \boldsymbol {\Sigma }^0_1(X) $
. Moreover,
$D^{\beta }_{\operatorname {CB}}(X)$
is discrete because
$||X||_{\operatorname {CB}} = \beta + 1$
, therefore
$A \cap D^{\beta }_{\operatorname {CB}}(X)$
is open in
$D^{\beta }_{\operatorname {CB}}(X)$
: let
$U \in \boldsymbol {\Sigma }^0_1(X)$
be such that
$U \cap D^{\beta }_{\operatorname {CB}}(X) = A \cap D^{\beta }_{\operatorname {CB}}(X)$
, so that the open subset
$ U_\beta = U \cup Y $
of
$ X $
satisfies
$A = U_\beta \setminus A'$
: then
$ A = D_{\beta +1}((U_\gamma )_{\gamma < \beta +1}) $
, witnessing
$A \in D_{\alpha }(\boldsymbol {\Sigma }^0_1(X))$
. If moreover X is simple, then
$D^{\beta }_{\operatorname {CB}}(X)=\{x\}$
is a singleton. If
$ x \notin A $
, then
$ A \subseteq Y $
and using the induction hypothesis we again obtain
$ A \in D_{\beta }(\boldsymbol {\Sigma }^0_1(Y))\subseteq D_{\beta }(\boldsymbol {\Sigma }^0_1(X)) $
. If instead
$ x \in A $
, then
$ A = X \setminus A' $
, hence
. In both cases,
.
Finally, suppose that
$\alpha $
is limit, and take a strictly increasing sequence
$(\alpha _n)_{n \in \omega }$
of nonzero ordinals cofinal in
$\alpha $
. For
$n\in \omega $
, consider the open sets
$V_n^* = X \setminus D^{\alpha _n}_{\operatorname {CB}}(X)$
. Apply the generalized reduction property to the sequence
$(V_n^*)_{n \in \omega }$
to obtain a clopen partition
$(V_n)_{n \in \omega }$
of
$ X $
such that
$V_n \subseteq V_n^*$
. Given any
$A\subseteq X$
, set
$A_n = A \cap V_n$
. Then
$A_n \subseteq V_n$
and
$||V_n||_{\operatorname {CB}} \le \alpha _n$
by Lemma 2.11, so by induction hypothesis
$A_n \in D_{\alpha _n}(\boldsymbol {\Sigma }^0_1(V_n)) \subseteq D_{\alpha _n}(\boldsymbol {\Sigma }^0_1(X))\subseteq D_{\alpha }(\boldsymbol {\Sigma }^0_1(X))$
. Write each
$ A_n $
as
$ A_n = D_{\alpha }((U^n_\beta )_{\beta <\alpha }) $
with
$ U_\beta ^n \in \boldsymbol {\Sigma }^0_1(X) $
. Then
$ A = \bigcup _{n \in \omega } (A_n \cap V_n) = D_{\alpha }((\bigcup _{n \in \omega } (U^n_\beta \cap V_n))_{\beta < \alpha })$
, hence
$A \in D_{\alpha }(\boldsymbol {\Sigma }^0_1(X))$
.
Notice that if
$ \boldsymbol {\Gamma } \in \mathrm {Class}_{\operatorname {NSD}}(X) $
and
$ \boldsymbol {\Gamma } \subseteq \boldsymbol {\Delta }^0_2(X) $
for a given zero-dimensional Polish space
$ X $
, then
$\boldsymbol {\Gamma } = D_{\beta }(\boldsymbol {\Sigma }^0_1(X))$
or
for some
$ \beta < \omega _1 $
. (Recall that by convention
$D_{0}(\boldsymbol {\Sigma }^0_1(X))=\{\emptyset \}$
.) Indeed, if
$ \rho \colon \mathcal {N} \to X $
is any retraction then
$ \rho ^{-1}(\boldsymbol {\Gamma }) \subseteq \boldsymbol {\Delta }^0_2(\mathcal {N}) $
is nonselfdual, and thus
$\rho ^{-1}(\boldsymbol {\Gamma }) = D_{\beta }(\boldsymbol {\Sigma }^0_1(\mathcal {N}))$
or
for some
$ \beta < \omega _1 $
by Wadge’s analysis of the first
$ \omega _1 $
-many nonselfdual pairs in
: since
$ \boldsymbol {\Gamma } = (\rho ^{-1}(\boldsymbol {\Gamma }))(X) $
by Lemma 3.4, we are done. This reproves [Reference Carroy, Medini and Müller11, Theorem 11.2], and since
is selfdual then Proposition 3.14 sharpens it by providing an upper bound on the possible
$\beta $
’s when
$ X $
is countable.
Corollary 3.15. Let X be a countable Polish space and
$\boldsymbol {\Gamma } \in \mathrm {Class}_{\operatorname {NSD}}(X)$
. Then
$\boldsymbol {\Gamma } = D_{\beta }(\boldsymbol {\Sigma }^0_1(X))$
or
for some
$\beta < ||X||_{\operatorname {CB}}$
.
We are now going to show that such bound is optimal.
Definition 3.16. Given
$\alpha <\omega _1$
, set
$$\begin{align*}A_\alpha =\{x\in K_{\alpha+1}\mid||x,K_{\alpha+1}||_{\operatorname{CB}}\mbox{ has parity opposite to }\alpha\}. \end{align*}$$
Proposition 3.17. For all
$\alpha <\omega _1$
,
.
Proof By definition,
$ A_0 = \emptyset $
and
$ K_{\alpha +1} \setminus A_\alpha = \operatorname {ptgl}( A_\beta ) $
if
$ \alpha = \beta +1 $
. More over, since when
$\alpha $
is limit the choice of the sequence
$ (\alpha _n)_{n \in \omega } $
in the definition of
$ K_{\alpha +1} = \operatorname {ptgl}((K_{\alpha _n+1})_{n\in \omega }) $
is irrelevant (up to homeomorphism), we can assume that the ordinals
$ \alpha _n $
are all odd, so that also in the limit case
$ K_{\alpha +1} \setminus A_\alpha = \operatorname {ptgl}((A_{\alpha _n})_{n \in \omega }) $
.
Recall the Wadge degrees
from (2.1). Since
$ D_{\alpha }(\boldsymbol {\Sigma }^0_1) \in \mathrm {Class}_{\operatorname {NSD}}(\mathcal {N}) $
, it is enough to prove by induction that
$ A_\alpha \equiv _{\mathsf {W}}^{K_{\alpha +1}, \mathcal {N}} B_\alpha $
for some/any
$ B_\alpha \in \mathbf {D}_\alpha $
. If
$ \alpha = 0 $
the result is obvious because
$ A_0 = \emptyset = B_0 $
. Suppose now that
$ \alpha = \beta +1 $
and fix
$ B_\beta \in \mathbf {D}_\beta $
. Let
$ f \colon K_{\beta +1} \to \mathcal {N} $
and
$ g \colon \mathcal {N} \to K_{\beta +1} $
witness
$ A_\beta \le _{\mathsf {W}}^{K_{\beta +1},\mathcal {N}} B_\beta $
and
$ B_\beta \le _{\mathsf {W}}^{\mathcal {N},K_{\beta +1}} A_\beta $
, respectively. Let
$ B_\alpha \in \mathbf {D}_\alpha $
be defined as in (2.1), that is,
$ B_\alpha = \{ (0)^n {^{\smallfrown }} (m+1) {^{\smallfrown }} x \mid n,m \in \omega \wedge x \in \neg B_\beta \} $
. Using
$ K_{\alpha +1} \setminus A_\alpha = \operatorname {ptgl}( A_\beta ) $
it is straightforward to see that
$ \operatorname {ptgl}(f) \colon K_{\alpha +1} \to \operatorname {ptgl}(\mathcal {N}) \subseteq \mathcal {N} $
witnesses
$ A_\alpha \le _{\mathsf {W}}^{K_{\alpha +1},\mathcal {N}} B_\alpha $
, and that the map
$ \operatorname {ptgl}(g) \circ \rho \colon \mathcal {N} \to K_{\alpha +1} $
, where
$ \rho \colon \mathcal {N} \to \operatorname {ptgl}(\mathcal {N}) $
is the retraction defined by
$ \rho ((0)^{\infty }) = (0)^{\infty } $
and
$ \rho ( (0)^n {^{\smallfrown }} (m+1) {^{\smallfrown }} x) = (0)^n {^{\smallfrown }} (1) {^{\smallfrown }} x $
, witnesses
$ B_\alpha \le _{\mathsf {W}}^{\mathcal {N},K_{\alpha +1}} A_\alpha $
. The limit case is similar.
As a by-product, we obtain a computation of
$||X||_{\operatorname {CB}} $
for countable spaces
$ X $
in terms of difference classes.
Proposition 3.18. Let X be a countable Polish space. Then
Proof By Proposition 3.14 we know that
for
$ \alpha = ||X||_{\operatorname {CB}} $
, so it is enough to show that
for every
$\alpha < ||X||_{\operatorname {CB}}$
.
Since
$\alpha +1\leq ||X||_{\operatorname {CB}}$
, by Proposition 3.12 we can suppose that
$K_{\alpha +1}$
is, up to homeomorphism, a closed subset of X. By Proposition 3.17 we get
. Fix a retraction
$ \rho \colon X \to K_{\alpha +1} $
and set
$ A^{\prime }_\alpha = \rho ^{-1}(A_\alpha ) $
: since
$ A_\alpha \equiv _{\mathsf {W}}^{K_{\alpha +1},X} A^{\prime }_\alpha $
, we get
, as desired.
Combining Corollary 3.15 and Proposition 3.18 we get a full description of nonselfdual Wadge classes in countable Polish spaces which nicely complements Theorem 2.9(2) and Lemma 3.13. Given any
$ \alpha < \Theta $
, we set
$ \mathrm {Class}_{\operatorname {NSD}}(\mathcal {N}) \restriction \alpha = \{ \boldsymbol {\Gamma } \in \mathrm {Class}_{\operatorname {NSD}}(\mathcal {N}) \mid ||\boldsymbol {\Gamma }||_{\mathcal {N}} < \alpha \} $
.
Corollary 3.19. Let
$ X $
be a countable Polish space. Then the nonselfdual Wadge pointclasses in
$ X $
are precisely
$ D_{\beta }(\boldsymbol {\Sigma }^0_1(X)) $
and their duals, for
$ \beta < ||X||_{\operatorname {CB}} $
.
In particular,
$ \mathrm {Class}_{\operatorname {NSD}}(X) = \{ \boldsymbol {\Gamma }(X) \mid \boldsymbol {\Gamma } \in \mathrm {Class}_{\operatorname {NSD}}(\mathcal {N}) \restriction 2 \cdot ||X||_{\operatorname {CB}} \} $
, and indeed
$ \langle \mathrm {Class}_{\operatorname {NSD}}(\mathcal {N}) \restriction 2 \cdot ||X||_{\operatorname {CB}} , {\subseteq } \rangle $
and
$ \langle \mathrm {Class}_{\operatorname {NSD}}(X), {\subseteq } \rangle $
are isomorphic via the map
$ \boldsymbol {\Gamma } \mapsto \boldsymbol {\Gamma }(X) $
.
4 Proof of Main Theorem 1
In view of the known facts from Propositions 2.2, 2.3, and 2.5, to completely describe
(up to isomorphism) for a given zero-dimensional Polish space
$ X $
we need to:
-
(1) determine which
$ 3 \leq \alpha < \Theta _X $
(if
$ \Theta _X> 3 $
) are selfdual and which are not, namely, determine the sets
$ \mathrm {Ord}_{\mathrm {SD}}(X) $
and
$ \mathrm {Ord}_{\mathrm {NSD}}(X) $
from Definition 2.4; -
(2) determine the value of
$ \Theta _X $
when
$ X $
is countable.
The first of these two goals splits into cases, depending on whether
$ \alpha $
is successor or not. For successor ordinals (Section 4.1), we will show that, as in the case of
$ \mathcal {N} $
and
$ \mathcal {C} $
, nonselfdual pairs and selfdual degrees alternate. This is relatively easy for uncountable spaces, although the proof necessarily differs from the one used in the case of
$ \mathcal {N} $
and
$ \mathcal {C} $
because we can no longer rely on the combinatorics of the space at hand, while in the countable case we need to perform a deeper analysis of the whole
and in particular of the maximal class(es).
For
$ \alpha $
limit (Section 4.2), we will immediately observe that for all
$ X $
, if
$ \alpha $
has uncountable cofinality then it is nonselfdual. The situation for limit ordinals of countable cofinality is more delicate: they can be selfdual, as it happens in
$ \mathcal {N} $
, or not, as in
$ \mathcal {C} $
. It turns out that for a general zero-dimensional Polish space
$ X $
the two behaviours (Baire-like and Cantor-like) can coexist, but in an orderly manner: if we denote by
$ \mathsf {Cof}_\omega $
the set of limit ordinals of countable cofinality, then
$ \mathrm {Ord}_{\mathrm {SD}}(X) \cap \mathsf {Cof}_\omega $
is always an initial segment of
$ \Theta _X\cap \mathsf {Cof}_\omega $
, which might be empty, have length
$ \Theta _X $
(i.e.,
$ \mathrm {Ord}_{\mathrm {NSD}}(X) \cap \mathsf {Cof}_\omega = \emptyset $
), or might be of the form
$ \alpha _X \cap \mathsf {Cof}_\omega $
for some countable ordinal
$ \alpha _X> \omega $
.
Finally, the deeper analysis of the (non)selfdual levels when
$ X $
is countable will also allow us to compute the value of
$ \Theta _X $
(Section 4.3), thus completing the proof of Main Theorem 1.
4.1 The alternating property
We say that a zero-dimensional Polish space X satisfies the alternating property if for all
$ \alpha \geq 1 $
such that
$\alpha + 1 < \Theta _X$
we have
$$\begin{align*}\alpha \in \mathrm{Ord}_{\mathrm{SD}}(X) \iff \alpha + 1 \in \mathrm{Ord}_{\mathrm{NSD}}(X). \end{align*}$$
The goal of this subsection is to prove that all zero-dimensional Polish spaces satisfy the alternating property. The following proposition shows that there cannot be two consecutive selfdual Wadge classes. In the case of
$ \mathcal {N} $
and
$ \mathcal {C} $
this is done constructively: there are explicit procedures to construct, starting from a given selfdual set, a nonselfdual pair immediately after it in the Wadge quasi-order. In the general case, instead, we cannot rely on any specific combinatorial property of the space at hand, thus we have to use a different and non-constructive proof.
Proposition 4.1 (
$ \mathsf {AD} $
).
Let X be a zero-dimensional Polish space, and
$A, A' \in \mathrm {Set}_{\operatorname {SD}}(X)$
be such that
$A <_{\mathsf {W}}^X A'$
. Then there is
$A" \in \mathrm {Set}_{\operatorname {NSD}}(X)$
satisfying
$A <_{\mathsf {W}}^X A" <_{\mathsf {W}}^X A'$
.
Proof Apply Proposition 2.6 to
$ A' $
to get a clopen partition
$ (V_n)_{n \in \omega } $
of
$ X $
and nonselfdual sets
$ (A_n)_{n \in \omega } $
such that
$ A_n <_{\mathsf {W}}^X A' $
and
$ A' = \bigcup _{n \in \omega } (A_n \cap V_n ) $
. If
$ A_n \le _{\mathsf {W}}^X A $
for all
$ n \in \omega $
via some continuous map
$ f_n \colon X \to X $
, then
$ f = \bigcup _{n \in \omega } (f_n \restriction V_n) $
would witness
$ A' \le _{\mathsf {W}}^X A $
, a contradiction. Since
$ A $
is selfdual, this means that by Wadge’s Lemma there is
$ \bar {n} \in \omega $
such that
$ A <_{\mathsf {W}}^X A_{\bar {n}} $
. Setting
$ A" = A_{\bar {n}} $
we are done.
In particular, Proposition 4.1 already yields one implication of the alternating property.
Corollary 4.2 (
$ \mathsf {AD} $
).
Let X be a zero-dimensional Polish space and
$\alpha \geq 1$
be such that
$ \alpha +1 < \Theta _X$
. If
$\alpha \in \mathrm {Ord}_{\mathrm {SD}}(X)$
, then
$\alpha + 1 \in \mathrm {Ord}_{\mathrm {NSD}}(X)$
.
The following corollary will be used in Proposition 4.5. To simplify the notation, let
be the coarse Wadge class associated with
$ D_{\beta }(\boldsymbol {\Sigma }^0_1(X)) $
.
Corollary 4.3. Let
$ X $
be a countableFootnote
11
Polish space. Then all sets
$ A \subseteq X $
such that
$ A \notin \bigcup _{\beta < ||X||_{\operatorname {CB}}} D_{\beta }^*(\boldsymbol {\Sigma }^0_1(X)) $
, if there exist any, are selfdual, Wadge equivalent to each other, and
.
Proof By Corollary 3.19, all sets
$ A $
as in the statement are selfdual. Assume towards a contradiction that there is
$ A' \notin \bigcup _{\beta < ||X||_{\operatorname {CB}}} D_{\beta }^*(\boldsymbol {\Sigma }^0_1(X)) $
such that
$ A' \not \le _{\mathsf {W}}^X A $
, so that
$ A' \in \mathrm {Set}_{\operatorname {SD}}(X) $
and
$ A <_{\mathsf {W}}^X A' $
. If
$ A' $
is
$ \le _{\mathsf {W}}^X $
-minimal with such properties, then we get a contradiction with (the restriction to Borel sets of) Proposition 4.1. Thus
$ A \equiv _{\mathsf {W}}^X A' $
for all
$ A,A' \notin \bigcup _{\beta < ||X||_{\operatorname {CB}}} D_{\beta }^*(\boldsymbol {\Sigma }^0_1(X)) $
.
Finally, fix
$ A $
as in the statement and consider an arbitrary
$ B \subseteq X $
. If
$ B \in D_{\beta }^*(\boldsymbol {\Sigma }^0_1(X)) $
for some
$ \beta < ||X||_{\operatorname {CB}}$
, then
$ B <_{\mathsf {W}}^X A $
by Wadge’s Lemma; otherwise,
$ B \equiv _{\mathsf {W}}^X A $
by the previous paragraph. In all cases
$ B \le _{\mathsf {W}}^X A $
, thus
.
Theorem 4.4 (
$ \mathsf {AD} $
).
All uncountable zero-dimensional Polish spaces satisfy the alternating property.
Proof Let X be an uncountable Polish zero-dimensional space, and consider any
$1 \leq \alpha < \Theta _X = \Theta $
. By Corollary 4.2 we only need to show that if
$\alpha \in \mathrm {Ord}_{\mathrm {NSD}}(X)$
, then
$\alpha + 1 \in \mathrm {Ord}_{\mathrm {SD}}(X)$
. By Theorem 2.9(1) there is
$ \boldsymbol {\Gamma } \in \mathrm {Class}_{\operatorname {NSD}}(\mathcal {N}) $
such that
$ \boldsymbol {\Gamma }(X) \in \mathrm {Class}_{\operatorname {NSD}}(X) $
and
$ ||\boldsymbol {\Gamma }(X)|| = \alpha $
. Since
$ X $
is uncountable then
$ \operatorname {ker}_{\operatorname {CB}}(X) $
is uncountable as well, thus we can partition
$ X $
into two uncountable clopen sets
$ X_0, X_1 $
. By Theorem 2.9(2) we have
$ \boldsymbol {\Gamma }(X_0) \in \mathrm {Class}_{\operatorname {NSD}}(X_0) $
and
. By Lemma 3.6 the smallest Wadge class containing
is selfdual, i.e.,
$ \alpha +1 \in \mathrm {Ord}_{\mathrm {SD}}(X) $
.
Combined with Theorem 2.9 and Lemma 3.4, the alternating property implies that the embedding from
into
induced by any retraction
$ \rho \colon \mathcal {N} \to X $
(Section 2.2) is almost surjective and might miss only limit levels of
. In particular, except for the countable cofinality case the rank of
$ \boldsymbol {\Gamma } \in \mathrm {Class}_{\mathsf {W}}(\mathcal {N}) $
and the rank (in
$ X $
) of
$ \boldsymbol {\Gamma }(X) $
are at distance at most
$1$
.
We now move to countable spaces.
Proposition 4.5. Let X be a countable Polish space. Then
is a Wadge class if and only if X is not simple.
Proof Suppose first that X is simple, so that in particular
$||X||_{\operatorname {CB}}=\alpha +1$
for some
$ \alpha < \omega _1 $
. By Corollary 3.19 we know that
, while the “moreover” part of Proposition 3.14 gives
. It follows that we cannot have
for
$ A \subseteq X $
, because if
$ B \subseteq X $
is complete for
$ D_{\alpha }(\boldsymbol {\Sigma }^0_1(X)) $
then
$ B, \neg B \le _{\mathsf {W}}^X A $
would imply
$ B <_{\mathsf {W}}^X A $
by
$ B \not \le _{\mathsf {W}}^X \neg B $
, a contradiction.
Suppose now that X is not simple. By Corollary 4.3 it is enough to show that there is some
$ A \subseteq X $
such that
$ A \notin \bigcup _{\beta <||X||_{\operatorname {CB}}}D_{\beta }^*(\boldsymbol {\Sigma }^0_1(X)) $
. We distinguish two cases.
If
$ ||X||_{\operatorname {CB}} = \lambda $
is limit, fix a strictly increasing sequence
$ (\alpha _i)_{i \in \omega } $
cofinal in
$ \lambda $
. By Lemma 2.10 we can find a clopen partition
$ (C_n)_{n \in \omega } $
of
$ X $
such that
$ ||C_n||_{\operatorname {CB}} < \lambda $
and
$ \sup _{n \in \omega } ||C_n||_{\operatorname {CB}}= \lambda $
. Fix an injection
$ i \mapsto n_i $
such that
$ ||C_{n_i}||_{\operatorname {CB}}> \alpha _i $
, so that
$ D_{\alpha _i}(\boldsymbol {\Sigma }^0_1(C_{n_i})) \in \mathrm {Class}_{\operatorname {NSD}}(C_{n_i}) $
for all
$ i \in \omega $
by Corollary 3.19. By Lemma 3.6 there is a selfdual Wadge class
$ A {\downarrow }_X $
which contains
$ \bigcup _{i \in \omega } D_{\alpha _i}(\boldsymbol {\Sigma }^0_1(X))$
, hence
$ A \notin \bigcup _{\beta < \lambda } D_{\beta }^*(\boldsymbol {\Sigma }^0_1(X)) $
because all the pointclasses
$ D_{\beta }(\boldsymbol {\Sigma }^0_1(X)) $
are distinct and nonselfdual by Corollary 3.19.
Suppose now that
$ \operatorname {\mathrm {tp}}(X) = (\beta +1, N) $
with
$ 1 < N \leq \omega $
. By Lemma 2.10 again we can find a clopen partition
$ (C_n)_{n < N} $
of
$ X $
such that
$ ||C_n||_{\operatorname {CB}} = \beta +1 $
, so that
$ \boldsymbol {\Gamma }_0(C_0) = D_{\beta }(\boldsymbol {\Sigma }^0_1(C_0)) \in \mathrm {Class}_{\operatorname {NSD}}(C_0) $
and
by Corollary 3.19. (For the remaining
$ 1 < n < N $
the choice of the nonselfdual pointclasses
$ \boldsymbol {\Gamma }_n(C_n) $
is irrelevant.) Then Lemma 3.6 allows us to conclude that there is a selfdual
$ A \subseteq X $
with
, which is again as required by Corollary 3.19.
Theorem 4.6. All countable zero-dimensional Polish spaces satisfy the alternating property.
Proof Let X be a countable Polish space. By Corollary 4.2, we only need to prove that if
$\alpha \in \mathrm {Ord}_{\mathrm {NSD}}(X)$
and
$\alpha +1<\Theta _X$
then
$\alpha +1\in \mathrm {Ord}_{\mathrm {SD}}(X)$
. By Corollary 3.19 there exists a
$\beta <||X||_{\operatorname {CB}}$
such that
$||D_{\beta }(\boldsymbol {\Sigma }^0_1(X))||=\alpha $
. We distinguish two cases.
Assume first that
$ \beta + 1 < ||X||_{\operatorname {CB}} $
. Then
$D^{\beta +1}_{\operatorname {CB}}(X) \neq \emptyset $
and
$D^{\beta }_{\operatorname {CB}}(X)$
is infinite, which implies that we can split X in two clopen sets
$X_0$
and
$X_1$
with
$||X_i||_{\operatorname {CB}} \geq \beta +1$
for
$i=0,1$
. (For example, pick
$ x \in D^{\beta }_{\operatorname {CB}}(X) \setminus D^{\beta +1}_{\operatorname {CB}}(X) $
and
$ X_0 $
clopen such that
$ X_0 \cap D^{\beta }_{\operatorname {CB}}(X) = \{ x \} $
: then
$ X_0 $
and
$ X_1 =X \setminus X_0 $
are as desired.) By Corollary 3.19 we have that
$\boldsymbol {\Gamma }_0(X_0) = D_{\beta }(\boldsymbol {\Sigma }^0_1(X_0)) \in \mathrm {Class}_{\operatorname {NSD}}(X_0) $
and
, so by Lemma 3.6 their supremum is selfdual and
$\alpha +1\in \mathrm {Ord}_{\mathrm {SD}}(X)$
.
Assume now
$ ||X||_{\operatorname {CB}} = \beta +1 $
. If
$ X $
were simple, then
$ \Theta _X = ||D_{\beta }(\boldsymbol {\Sigma }^0_1(X))|| +1 = \alpha +1 $
by Proposition 3.14, contradicting the choice of
$\alpha $
. Thus
$ X $
is not simple and Proposition 4.5 together with Corollary 4.3 show that at level
$ \alpha + 1 $
we have the selfdual Wadge class
, so
$ \alpha +1 \in \mathrm {Ord}_{\mathrm {SD}}(X) $
again and we are done.
4.2 Limit levels
We now move to the analysis of limit levels of
. We first settle the uncountable cofinality case.
Proposition 4.7 (
$ \mathsf {AD} $
).
Let
$ X $
be a zero-dimensional Polish space. If
$1 \leq \alpha < \Theta _X$
is a limit ordinal and
$\alpha \in \mathrm {Ord}_{\mathrm {SD}}(X)$
, then
$ \operatorname {cof}(\alpha )= \omega $
. Thus at limit levels of uncountable cofinality there is always a nonselfdual pair.
Proof Let
$ A \in \mathrm {Set}_{\operatorname {SD}}(X) $
be such that
$ ||A||_{\mathsf {W}}^X = \alpha $
. By Proposition 2.6, there are sets
$ (A_n)_{n \in \omega } $
such that
$ \alpha = \sup _{n \in \omega } (||A_n||_{\mathsf {W}}^X+1) $
: this already shows that
$ \alpha $
has countable cofinality.
To deal with the countable cofinality case, we need to distinguish whether
$ \operatorname {ker}_{\operatorname {CB}}(X) $
is compact or not. We begin with a technical lemma, which might be of independent interest because it shows that
$ \omega \times \mathcal {C} $
is minimal for closed embeddability among zero-dimensional Polish spaces with non-compact perfect kernel.
Lemma 4.8. Let X be a zero-dimensional Polish space. The following are equivalent:
-
(i)
$\operatorname {ker}_{\operatorname {CB}}(X)$
is not compact
$;$
-
(ii) there exists an infinite (countable) clopen partition
$ (C_n)_{n \in \omega } $
of X such that each
$ C_n $
is uncountable
$;$
-
(iii)
$ X $
contains a closed set
$ F $
homeomorphic to
$ \omega \times \mathcal {C} $
.
Proof If
$ X $
is countable, then all of (i)–(iii) are false, so without loss of generality we may assume that
$ X $
is uncountable.
We start with (i)
$ \Rightarrow $
(ii). Since
$ \operatorname {ker}_{\operatorname {CB}}(X) $
is not compact, by Fact 3.8 there is an infinite clopen partition
$(C^{\prime }_n)_{n \in \omega }$
of
$\operatorname {ker}_{\operatorname {CB}}(Z)$
, and since
$ \operatorname {ker}_{\operatorname {CB}}(X) $
is perfect each
$ C^{\prime }_n $
is uncountable. Thus it is enough to set
$ C_n = \rho ^{-1}(C^{\prime }_n) $
where
$ \rho \colon X \to \operatorname {ker}_{\operatorname {CB}}(X) $
is any retraction.
Assume now that (ii) holds. Since the
$ C_n $
’s are uncountable Polish spaces, they contain a closed set
$ F_n \subseteq C_n $
homeomorphic to
$ \mathcal {C} $
: then
$ F = \bigcup _{n \in \omega } F_n $
is still closed and homeomorphic to
$ \omega \times \mathcal {C} $
, hence (iii) holds.
Finally, we prove (iii)
$ \Rightarrow $
(i). Since
$ F $
is perfect, then
$ F \subseteq \operatorname {ker}_{\operatorname {CB}}(X) $
. But then
$ \operatorname {ker}_{\operatorname {CB}}(X) $
cannot be compact, otherwise so would be
$ F $
and
$ \omega \times \mathcal {C} $
, which is false.
Theorem 4.9 (
$ \mathsf {AD} $
).
Let X be an uncountable zero-dimensional Polish space, so that
$ \Theta _X = \Theta $
. Then the following are equivalent:
-
(i)
$ \operatorname {ker}_{\operatorname {CB}}(X) $
is not compact
$;$
-
(ii)
$ \alpha \in \mathrm {Ord}_{\mathrm {SD}}(X) $
for every limit
$ \alpha < \omega _1 ;$
-
(iii) if
$ \alpha \in \Theta _X \cap \mathsf {Cof}_\omega $
, then
$ \alpha \in \mathrm {Ord}_{\mathrm {SD}}(X) $
.
Proof It is enough to prove (i)
$ \Rightarrow $
(iii) and (the contrapositive of) (ii)
$ \Rightarrow $
(i).
Assume first that
$ \operatorname {ker}_{\operatorname {CB}}(X) $
is not compact, so that by Lemma 4.8 there is an infinite clopen partition
$ (C_n)_{n \in \omega } $
of
$ X $
into uncountable pieces. Given any
$ \alpha \in \Theta _X \in \mathsf {Cof}_\omega $
, pick a strictly increasing sequence of ordinals
$ (\alpha _n)_{n \in \omega } $
cofinal in
$\alpha $
such that
$ \alpha _n \in \mathrm {Ord}_{\mathrm {NSD}}(X) $
for all
$ n \in \omega $
(which exists by the alternating property), and using Theorem 2.9(1) let
$ \boldsymbol {\Gamma }_n \in \mathrm {Class}_{\operatorname {NSD}}(\mathcal {N}) $
be such that
$ ||\boldsymbol {\Gamma }_n(X)||_X = \alpha _n $
. By Theorem 2.9(2) and uncountability of
$ C_n $
, we have
$ \boldsymbol {\Gamma }_n(C_n) \in \mathrm {Class}_{\operatorname {NSD}}(C_n) $
as well, hence by Lemma 3.6 the supremum
$ \boldsymbol {\Gamma } \in \mathrm {Class}_{\mathsf {W}}(X) $
of the pointclasses
$ \boldsymbol {\Gamma }_n(X) $
is selfdual, i.e.,
$ \alpha \in \mathrm {Ord}_{\mathrm {SD}}(X) $
.
Assume now that
$ \operatorname {ker}_{\operatorname {CB}}(X) $
is compact and let
$ \beta = ||X||_{\operatorname {CB}} $
. Consider any
$ \beta + \omega < \alpha < \omega _1 $
: we want to show that if
$ \alpha \in \mathrm {Ord}_{\mathrm {SD}}(X) $
then
$ \alpha $
is a successor ordinal, so that any limit ordinal between
$ \beta +\omega $
and
$ \omega _1 $
witnesses the failure of (ii). Let
$ A \subseteq X $
be a selfdual set such that
$ ||A||_{\mathsf {W}}^X = \alpha $
. By Proposition 2.6 there exists a clopen partition
$(V_n)_{n \in \omega }$
of
$ X $
and sets
$(A_n)_{n \in \omega }$
such that
$A_n <_{\mathsf {W}}^X A = \bigcup _{n \in \omega } (A_n \cap V_n)$
and
$ \alpha = \sup _{n \in \omega }(\alpha _n+1) $
for
$ \alpha _n= ||A_n||_{\mathsf {W}}^X $
. Recall also from Remark 2.7(1) that we can assume that for each
$ n \in \omega $
either
$ A_n = X $
, or else
$ A_n \subseteq V_n $
. Since
$ \operatorname {ker}_{\operatorname {CB}}(X) $
is compact, there is
$ N \in \omega $
such that
$ V_n \cap \operatorname {ker}_{\operatorname {CB}}(X) = \emptyset $
for all
$ n \geq N $
. Fix any
$ n \geq N $
. Then
$ ||V_n||_{\operatorname {CB}} \leq \beta $
by Lemma 2.11, hence if
$ A_n \subseteq V_n $
then
$ A_n \in D_{\beta }(\boldsymbol {\Sigma }^0_1(V_n)) \subseteq D_{\beta }(\boldsymbol {\Sigma }^0_1(X)) $
by Proposition 3.14, which means that
$ \alpha _n < \beta +\omega $
because by Theorem 2.9 and the alternating property
$ ||D_{\beta }(\boldsymbol {\Sigma }^0_1(X))||_X \leq ||D_{\beta }(\boldsymbol {\Sigma }^0_1(\mathcal {N}))||_{\mathcal {N}} = 2 \beta +1 $
. If instead
$ A_n = X $
, then
$ \alpha _n = 1 $
. Hence
$ \alpha _n + 1 < \beta +\omega $
for all
$ n \geq N $
. Since
$ \alpha> \beta +\omega $
, this means that necessarily
$ \alpha = \sup _{n \in \omega } (\alpha _n+1) = \sup _{n < N } (\alpha _n+1) = \max _{n < N} (\alpha _n+1) $
, and thus
$\alpha $
is a successor ordinal.
If
$ X $
is countable, so that
$ \operatorname {ker}_{\operatorname {CB}}(X) = \emptyset $
is trivially compact, then
$ \Theta _X < \omega _1 < \Theta = \Theta _{\mathcal {N}} $
(see Section 4.3 for the exact computation of
$ \Theta _X $
), and thus
cannot be isomorphic to
. Combining this with Theorem 4.9 and our previous results (and recalling that, up to homeomorphism, any zero-dimensional Polish space can be construed as a closed subspace of
$ \mathcal {N} $
) we get:
Corollary 4.10 (
$ \mathsf {AD} $
).
If
$ X $
is a zero-dimensional Polish space, then
is order-isomorphic to
if and only if
$ \operatorname {ker}_{\operatorname {CB}}(X) $
is not compact, where (when it exists) the isomorphism is canonicallyFootnote
12
induced by
$ A \mapsto \rho ^{-1}(A) $
for some/any retraction
$ \rho \colon \mathcal {N} \to X $
.
We now move to spaces
$ X $
with a compact perfect kernel (this includes the case of countable spaces). It turns out that the compact rank from Section 3.3 provides a crucial dividing line.
Theorem 4.11 (
$ \mathsf {AD} $
).
Let X be a zero-dimensional Polish space such that
$ \operatorname {ker}_{\operatorname {CB}}(X) $
is compact, and let
$\alpha \in \Theta _X \cap \mathsf {Cof}_\omega $
.
-
(1) If
$\alpha> ||X||_{\operatorname {Comp}}$
, then
$\alpha \in \mathrm {Ord}_{\mathrm {NSD}}(X)$
. -
(2) If
$\alpha < ||X||_{\operatorname {Comp}}$
, then
$\alpha \in \mathrm {Ord}_{\mathrm {SD}}(X)$
. -
(3) If
$ \alpha = ||X||_{\operatorname {Comp}} $
, then
$ \alpha \in \mathrm {Ord}_{\mathrm {SD}}(X) $
if and only if there is a clopen partition
$ (V_n)_{n \in \omega } $
of
$ X $
and
$ N \in \omega $
such that
$ \alpha = \sup _{n \geq N} (||V_n||_{\operatorname {CB}}+1) $
.
Proof (1) Suppose towards a contradiction that
$ \alpha> ||X||_{\operatorname {Comp}} $
but
$ \alpha \in \mathrm {Ord}_{\mathrm {SD}}(X) $
: we adapt the proof of Theorem 4.9 to show that
$\alpha $
needs to be a successor ordinal, contradicting
$ \alpha \in \mathsf {Cof}_\omega $
. Given any
$ A \in \mathrm {Set}_{\operatorname {SD}}(X) $
such that
$ ||A||_{\mathsf {W}}^X = \alpha $
, use Proposition 2.6 to get a clopen partition
$(V_n)_{n \in \omega }$
of
$ X $
and sets
$(A_n)_{n \in \omega }$
such that
$A_n <_{\mathsf {W}}^X A = \bigcup _{n \in \omega } (A_n \cap V_n)$
and
$ \alpha = \sup _{n \in \omega } (\alpha _n+1) $
for
$ \alpha _n= ||A_n||_{\mathsf {W}}^X $
. Since by definition
$D^{||X||_{\operatorname {Comp}}}_{\operatorname {CB}}(X)$
is compact, there exists
$N \in \omega $
such that
$V_n \subseteq X \setminus D^{||X||_{\operatorname {Comp}}}_{\operatorname {CB}}(X)$
for all
$n \geq N$
. Using that either
$ A_n = X $
or
$ A_n \subseteq V_n $
, this implies that
$ A_n \in D_{||X||_{\operatorname {Comp}}}(\boldsymbol {\Sigma }^0_1(X)) $
for all
$ n \geq N $
. Arguing as in Theorem 4.9 (and using that
$ \alpha> ||X||_{\operatorname {Comp}} $
is limit), we get that
$ ||D_{||X||_{\operatorname {Comp}}}(\boldsymbol {\Sigma }^0_1(X))||_X \leq ||D_{||X||_{\operatorname {Comp}}}(\boldsymbol {\Sigma }^0_1(\mathcal {N}))||_{\mathcal {N}} = 2 \cdot ||X||_{\operatorname {Comp}}+1 < \alpha $
, hence
$ \sup _{n \geq N} (\alpha _n+1) < \alpha $
. Thus
$ \alpha = \sup _{n \in \omega }(\alpha _n+1) = \max _{n < N} (\alpha _n+1) $
is a successor ordinal, a contradiction.
(2) Suppose now that
$\alpha <||X||_{\operatorname {Comp}}$
, so that in particular X is not compact. Then by Lemma 3.10 there is a clopen partition
$(C_n)_n$
of X such that
$D^{\alpha +1}_{\operatorname {CB}}(X)\subseteq C_0$
and
$||C_{n+1}||_{\operatorname {CB}}=\alpha +1$
for all
$n\in \omega $
. For every
$ n \geq 1 $
and
$ \beta < \alpha $
, Corollary 3.19 implies that
$ D_{\beta }(\boldsymbol {\Sigma }^0_{1}(C_n)) \in \mathrm {Class}_{\operatorname {NSD}}(C_n) $
. Moreover,
$ \beta \leq ||D_{\beta }(\boldsymbol {\Sigma }^0_{1}(X))|| \leq 2\beta +1 $
: if
$ X $
is countable this follows from the alternating property and Corollary 3.19 together with
$ ||X||_{\operatorname {CB}} \geq ||X||_{\operatorname {Comp}}> \alpha > \beta $
, while if
$ X $
is uncountable we can use the alternating property and Lemma 3.13. Since
$\alpha $
is limit, then
$ 2\beta +1 < \alpha $
and hence
$ \beta \leq ||D_{\beta }(\boldsymbol {\Sigma }^0_{1}(X))|| < \alpha $
. Therefore taking any strictly increasing sequence
$ (\alpha _n)_{n \geq 1} $
cofinal in
$ \alpha $
we have
$ D_{\alpha _n}(\boldsymbol {\Sigma }^0_1(C_n)) \in \mathrm {Class}_{\operatorname {NSD}}(C_n) $
,
$ ||D_{\alpha _n}(\boldsymbol {\Sigma }^0_{1}(X))||_{X} < \alpha $
, and
$ \sup _{n \geq 1} ||D_{\alpha _n}(\boldsymbol {\Sigma }^0_{1}(X))|| = \alpha $
. Setting
$ \boldsymbol {\Gamma }_0 = \{ \mathcal {N} \} $
and
$ \boldsymbol {\Gamma }_n = D_{\alpha _n}(\boldsymbol {\Sigma }^0_1) $
for
$ n \geq 1 $
, and applying Lemma 3.6, we then get
$ \alpha \in \mathrm {Ord}_{\mathrm {SD}}(X) $
, as desired.
(3) The proof of the desired equivalence is a variation of the arguments for parts (1) and (2). Assume first that
$ \alpha \in \mathrm {Ord}_{\mathrm {SD}}(X) $
. Given any
$ A \in \mathrm {Set}_{\operatorname {SD}}(X) $
such that
$ ||A||_{\mathsf {W}}^X=\alpha $
, let
$ (V_n)_{n \in \omega } $
and
$ (A_n)_{n \in \omega } $
be as in Proposition 2.6. Since
$ D^{\alpha }_{\operatorname {CB}}(X) $
is compact, there is
$ N \in \omega $
such that
$ V_n \cap D^{\alpha }_{\operatorname {CB}}(X) = \emptyset $
for all
$ n \geq N $
, which also means
$ ||V_n||_{\operatorname {CB}} \leq \alpha $
by Lemma 2.11. If
$ ||V_n||_{\operatorname {CB}} = \alpha $
for some
$ n \geq N $
, then we apply Lemma 2.10 to all such countable spaces
$ V_n $
and get the desired partition. (Recall that
$\alpha $
is limit.) So we can assume that
$ ||V_n||_{\operatorname {CB}} + 1 \leq \alpha $
for all
$ n \geq N $
and we only need to show that for every
$ \beta < \alpha $
there is
$ n \geq N $
with
$ ||V_n||_{\operatorname {CB}} \geq \beta $
. Assume not: since we can assume that either
$ A_n = X $
or
$ A_n \subseteq V_n $
, by Proposition 3.14 we would have
$ A_n \in D_{||V_n||_{\operatorname {CB}}}(\boldsymbol {\Sigma }^0_1(V_n)) \subseteq D_{\beta }(\boldsymbol {\Sigma }^0_1(V_n)) \subseteq D_{\beta }(\boldsymbol {\Sigma }^0_1(X)) $
for all
$ n \geq N $
, and since the usual computation gives
$ ||D_{\beta }(\boldsymbol {\Sigma }^0_1(X))||_X \leq 2 \beta +1 $
and
$ 2 \beta +1 < \alpha $
because
$\alpha $
is limit, then
$ \alpha = \sup _{n \in \omega } (||A_n||_{\mathsf {W}}^X+1) = \max _{n < N} (||A_n||_{\mathsf {W}}^X+1) $
, contradicting the fact that
$ \alpha $
is limit.
Conversely, let
$ (V_n)_{n \in \omega } $
and
$ N \in \omega $
be as in part (3), and fix a sequence
$ (\alpha _k)_{k \in \omega } $
cofinal in
$ \alpha $
. Since
$\alpha $
is limit, the assumption on
$ (V_n)_{n \in \omega } $
implies that for each
$ k \in \omega $
we can find
$ n_k \geq N $
such that
$ \alpha _k < ||V_{n_k}||_{\operatorname {CB}} $
and further ensure that the map
$ k \mapsto n_k $
is injective. By Corollary 3.19 the class
$ \boldsymbol {\Gamma }_{n_k}(V_{n_k}) = D_{\alpha _k}(\boldsymbol {\Sigma }^0_1(V_{n_k})) $
is nonselfdual in
$ V_{n_k} $
, and as in the proof of part (2) one can show that
$ \alpha _{k} \leq ||D_{\alpha _k}(\boldsymbol {\Sigma }^0_1(X))|| \leq 2 \alpha _k +1 < \alpha $
. Setting
$ \boldsymbol {\Gamma }_n = \{ \mathcal {N} \} $
for all
$ n \in \omega $
which are not of the form
$ n_k $
and applying Lemma 3.6 we get
$ \alpha \in \mathrm {Ord}_{\mathrm {SD}}(X) $
, as desired.
Remark 4.12. Together with Lemma 2.10, Theorem 4.11(3) shows that if
$ X $
is a countable zero-dimensional Polish space and
$ ||X||_{\operatorname {CB}} = ||X||_{\operatorname {Comp}} = \alpha < \Theta _X $
is limit, then necessarily
$ \alpha \in \mathrm {Ord}_{\mathrm {SD}}(X) $
.
Concerning the characterization in Theorem 4.11(3), it is easy to provide examples of both sorts.
Example 4.13. Let
$ \alpha $
be a countable limit ordinal and fix a strictly increasing sequence
$ (\alpha _n)_{n \in \omega } $
cofinal in
$ \alpha $
. Denote by
$ \bigoplus _{n \in \omega } X_n $
the (disjoint) sum of the topological spaces
$ X_n $
. (If
$ X_n \subseteq \mathcal {N} $
, then up to homeomorphism the sum can be construed as
$ \bigoplus _{n \in \omega } X_n= \bigcup _{n \in \omega } \{ n {^{\smallfrown }} x \mid n \in \omega \wedge x \in X_n \} $
.) Then the spaces
$$\begin{align*}Y_\alpha = \bigoplus_{n \in \omega} K_{\alpha_n+1} \qquad \text{and} \qquad Z_\alpha = \operatorname{ptgl}((\omega \times K_{\alpha_n+1})_{n \in \omega}) \end{align*}$$
are both countable zero-dimensional Polish spaces with compact rank
$ \alpha $
, and by Theorem 4.11(3) we have
$ \alpha \in \mathrm {Ord}_{\mathrm {SD}}(Y_\alpha ) $
but
$ \alpha \in \mathrm {Ord}_{\mathrm {NSD}}(Z_\alpha ) $
. Moreover,
$ ||Y_\alpha ||_{\operatorname {CB}} = \alpha $
and
$ ||Z_\alpha ||_{\operatorname {CB}} = \alpha +1 $
(indeed,
$ \operatorname {\mathrm {tp}}(Z_\alpha ) = (\alpha +1,1) $
, i.e.,
$ Z_\alpha $
is also simple), and the values for such CB-ranks are as small as possible by Remark 4.12.
The following result, together with Proposition 4.7 and Theorem 4.9, completes the proof of part (3) of Main Theorem 1. Let
$ \alpha _X $
be the smallest
$ \alpha _X \leq \Theta _X $
such that
$ \alpha \in \mathrm {Ord}_{\mathrm {SD}}(X) $
for all limit
$ \alpha < \alpha _X $
with
$ \operatorname {cof}(\alpha )= \omega $
. By Theorems 4.9 and 4.11 we get:
Corollary 4.14 (
$ \mathsf {AD} $
).
Let
$ X $
be an arbitrary zero-dimensional Polish space. Then either
$ \alpha _X = \Theta _X = \Theta $
(if
$ \operatorname {ker}_{\operatorname {CB}}(X) $
is not compact), or else
$ \alpha _X < \omega _1 $
and
$ \alpha \in \mathrm {Ord}_{\mathrm {NSD}}(X) $
for all limit
$ \alpha _X \leq \alpha < \Theta _X $
.
Indeed, by Theorems 4.9 and 4.11 the ordinal
$ \alpha _X $
can be computed as follows:
-
• If
$ \operatorname {ker}_{\operatorname {CB}}(X) $
is not compact, then
$ X $
is uncountable and
$ \alpha _X = \Theta _X = \Theta $
. -
• If
$ \operatorname {ker}_{\operatorname {CB}}(X) $
is compact and
$ ||X||_{\operatorname {Comp}} $
is a successor ordinal, then
$ ||X||_{\operatorname {Comp}} = \lambda + n $
for some limit or null
$ \lambda < \omega _1 $
and
$ n \in \omega \setminus \{ 0 \} $
: set
$ \alpha _X = \lambda +1 $
if
$ \lambda> 0 $
and
$ \alpha _X =0 $
otherwise. -
• If
$ \operatorname {ker}_{\operatorname {CB}}(X) $
is compact and
$ ||X||_{\operatorname {Comp}} = \lambda $
is limit, then we have to distinguish cases depending on whether
$ ||X||_{\operatorname {Comp}} \in \mathrm {Ord}_{\mathrm {SD}}(X) $
or not. In the former case, we again set
$ \alpha _X = \lambda +1 $
. Otherwise we set
$ \alpha _X = 0 $
if
$ \lambda = \omega $
,
$ \alpha _X = \lambda '+1 $
if
$ \lambda $
is of the form
$ \lambda '+ \omega $
for some limit
$ \lambda '< \omega _1 $
, and
$ \alpha _X = \lambda $
otherwise (i.e., if
$ \lambda $
is a limit of limit ordinals).
In particular,
$ \alpha _X $
is either
$ 0 $
, or the successor of a countable limit ordinal, or a limit of countable limit ordinals, or the ordinal
$ \Theta $
. Moreover, when
$ \alpha _X < \omega _1 $
we have
$ \alpha _X \leq ||X||_{\operatorname {CB}} $
unless
$ ||X||_{\operatorname {CB}} $
is limit, in which case we might have
$ \alpha _X \leq ||X||_{\operatorname {CB}} $
or
$ \alpha _X = ||X||_{\operatorname {CB}}+1 $
.
4.3 Length of the Wadge hierarchy for countable spaces
By Proposition 2.5 we know that if a zero-dimensional Polish space
$ X $
is uncountable, then
$ \Theta _X = \Theta $
. Thus to complete the proof of part (4) of Main Theorem 1 we just need to compute the value of
$ \Theta _X $
when
$ X $
is countable. Notice that since we consider only nonempty spaces
$ X $
, the minimal possible length for
is
$ 2 $
.
Theorem 4.15. Let X be a countable Polish space. Then
$\Theta _X = 2 \cdot ||X||_{\operatorname {CB}} + \varepsilon _X$
, for some
$\varepsilon _X \in \{-1,0,1\}$
.
Proof Let
$||X||_{\operatorname {CB}} = \lambda + n$
for some
$\lambda $
limit or null and
$n\in \omega $
. We distinguish various cases.
If
$ \lambda = 0 $
(i.e.,
$ ||X||_{\operatorname {CB}} $
is finite), then
$ n> 0 $
because
$ X \neq \emptyset $
. Then by Corollary 3.19 the largest nonselfdual Wadge classes in
are
$ D_{n-1}(\boldsymbol {\Sigma }^0_1(X)) $
and its dual, and by the alternating property (and the fact that the Wadge rank starts by
$ 1 $
) we have
$ ||D_{n-1}(\boldsymbol {\Sigma }^0_1(X))|| = 2n-1 $
. By Proposition 3.14 and Proposition 4.5, it follows that
$ \Theta _X = 2n = 2 \cdot ||X||_{\operatorname {CB}} $
if
$ X $
is simple, in which case we set
$ \varepsilon _X = 0 $
, and
$ \Theta _X = 2n +1= 2 \cdot ||X||_{\operatorname {CB}}+1 $
otherwise, in which case
$ \varepsilon _X = 1 $
.
Assume now that
$ \lambda> 0 $
. By the alternating property and Corollary 3.19 we know that
$ \Theta _X \geq \lambda $
and that the ranks of difference classes of the form
$ D_{\beta }(\boldsymbol {\Sigma }^0_1(X)) $
with
$ \beta < \lambda $
are cofinal in
$ \lambda $
. So we have two subcases. If
$ n = 0 $
, then
$ X $
is not simple and Proposition 4.5 gives that
is a selfdual Wadge class of rank exactly
$\lambda $
: thus
$ \Theta _X = \lambda +1 = 2 \cdot ||X||_{\operatorname {CB}}+1 $
and we set
$ \varepsilon _X = 1 $
. If instead
$ n> 0 $
, then by the alternating property and Corollary 3.19 again we know that there are precisely
$ n $
-many nonselfdual Wadge classes with rank
$ \geq \lambda $
, i.e., the classes
$ D_{\lambda +i}(\boldsymbol {\Sigma }^0_1(X)) $
for
$ i < n $
, which by the alternating property means that we have at least
$ 2n-1 $
Wadge classes with rank
$ \geq \lambda $
. Then we have to consider two issues. First, if
$ \alpha _X> \lambda $
then
$ \lambda \in \mathrm {Ord}_{\mathrm {SD}}(X) $
and
$ ||D_{\lambda }(\boldsymbol {\Sigma }^0_1(X))|| = \lambda +1$
, i.e., we have one more (selfdual) class before
$ D_{\lambda }(\boldsymbol {\Sigma }^0_1(X)) $
; if instead
$ \alpha _X \leq \lambda $
then
$ ||D_{\lambda }(\boldsymbol {\Sigma }^0_1(X))|| = \lambda $
and there is no additional class to be taken into account. Second, by Proposition 3.14 and Proposition 4.5 we know that if
$ X $
is simple then
$ D_{\lambda +n-1}(\boldsymbol {\Sigma }^0_1(X)) $
is the last Wadge class in
, whereas if
$ X $
is not simple then
forms yet one more Wadge class immediately after
$ D_{\lambda +n-1}(\boldsymbol {\Sigma }^0_1(X)) $
and its dual. Summing up the discussion above we have that
$ \Theta _X = 2 \cdot ||X||_{\operatorname {CB}} + \varepsilon _X $
where
$ \varepsilon _X = \beta _X + \delta _X -1 $
and
$ \beta _X = 0 $
if
$ X $
is simple and
$ \beta _X = 1 $
otherwise, while
$ \delta _X = 0 $
if
$ \alpha _X \leq \lambda $
and
$ \delta _X = 1 $
otherwise.
4.4 Optimality
The previous results are sharp, meaning that all possible shapes for a Wadge hierarchy
which are coherent with our description are actually realized by some zero-dimensional Polish space
$ X $
. We actually prove a finer result, showing that all possible configurations of the parameters
$ ||X||_{\operatorname {CB}} $
,
$ ||X||_{\operatorname {Comp}} $
, and, if
$ X $
is countable,
$ \operatorname {\mathrm {tp}}(X) $
can be realized in suitable spaces, together with some additional features relevant to our analysis of
. The limitations in the statement follow from the possible values of the CB-rank and the CB-type (and the fact that we consider only nonempty spaces), the definition of the compact rank, Lemma 2.10, and Remark 4.12. Notice also that the additional conditions on
$ \beta $
and
$ \gamma $
are relevant only in part (1): in particular, in part (2) the ordinal
$ \gamma $
can unconditionally assume any value
$ \leq \alpha $
.
Proposition 4.16. Let
$ (\alpha ,\beta ) \in \omega _1 \times (\omega +1) $
and
$ \gamma \leq \alpha $
. Further assume that
$ \beta \geq 1 $
,
$ \beta = \omega $
if
$ \alpha $
is limit,
$ \gamma = \alpha $
if
$ \beta = \omega $
, and
$ \gamma \leq \alpha ' $
if
$ \alpha = \alpha '+1 $
and
$ \beta < \omega $
.
-
(1) If
$ \alpha \geq 1 $
there is a countable (zero-dimensional) Polish space
$X $
with
$ \operatorname {\mathrm {tp}}(X) = (\alpha ,\beta ) $
and
$ ||X||_{\operatorname {Comp}} = \gamma $
. Moreover, when
$ \gamma $
is limit we can require to have
$ \gamma \in \mathrm {Ord}_{\mathrm {SD}}(X) $
, or else to have
$ \gamma \in \mathrm {Ord}_{\mathrm {NSD}}(X) $
(provided that
$ \alpha> \gamma $
, in which case
$\alpha $
needs to be a successor ordinal). -
(2) There is an uncountable zero-dimensional Polish space
$ X $
with
$ \operatorname {ker}_{\operatorname {CB}}(X) $
compact,
$ ||X||_{\operatorname {CB}} =\alpha $
, and
$ ||X||_{\operatorname {Comp}} = \gamma $
. Moreover, when
$ \gamma $
is limit we can freely decide to have
$ \gamma \in \mathrm {Ord}_{\mathrm {SD}}(X) $
or
$ \gamma \in \mathrm {Ord}_{\mathrm {NSD}}(X) $
. -
(3) There is a(n uncountable) zero-dimensional Polish space
$ X $
with
$ \operatorname {ker}_{\operatorname {CB}}(X) $
non-compact and
$ ||X||_{\operatorname {CB}} = \alpha $
.
Proof (1) Further developing Example 4.13, for any
$ \delta < \omega _1 $
we set
$$\begin{align*}Y_\delta = \bigoplus_{n \in \omega} K_{\delta_n+1} \qquad \text{and} \qquad Z_\delta = \operatorname{ptgl}((\omega \times K_{\delta_n+1})_{n \in \omega}), \end{align*}$$
where
$ (\delta _n)_{n \in \omega } $
is a sequence cofinal in
$ \delta $
if the latter is limit, or the constant sequence with value
$ \delta ' $
if
$ \delta = \delta '+1 $
. Then
$ \operatorname {\mathrm {tp}}(Y_\delta ) = (\delta ,\omega ) $
,
$ \operatorname {\mathrm {tp}}(Z_\delta ) = (\delta +1,1) $
, and
$ ||Y_\delta ||_{\operatorname {Comp}} = ||Z_\delta ||_{\operatorname {Comp}} = \delta $
. Moreover, if
$ \delta $
is limit then
$ \delta \in \mathrm {Ord}_{\mathrm {SD}}(Y_\delta ) $
while
$ \delta \in \mathrm {Ord}_{\mathrm {NSD}}(Z_\delta ) $
.
The space
$ X = Y_\alpha $
already settles the case when
$ \alpha $
is limit or when
$\alpha $
is a successor but
$ \beta = \omega $
. So assume that
$ \alpha = \alpha ' +1 $
and
$ 1 \leq \beta < \omega $
, in which case
$ \gamma \leq \alpha ' $
. If
$ \gamma = \alpha ' $
then
$ X = (\beta \times K_\alpha ) \oplus Y_{\alpha '} $
has CB-type
$ (\alpha ,\beta ) $
and compact rank
$ \gamma $
, as required. For the additional part, notice that if
$ \alpha ' = \gamma $
is limit, this construction gives
$ \gamma \in \mathrm {Ord}_{\mathrm {SD}}(X) $
; if instead we want
$\gamma \in \mathrm {Ord}_{\mathrm {NSD}}(X) $
, then we use
$ X = \beta \times Z_{\alpha '} $
, which has still CB-type
$ (\alpha ,\beta ) $
and compact rank
$ \gamma $
but is such that
$ \gamma \in \mathrm {Ord}_{\mathrm {NSD}}(X) $
, as desired. The remaining cases
$ \gamma < \alpha ' $
are treated similarly: we set
$ X = (\beta \times K_\alpha ) \oplus Y_{\gamma } $
if
$ \gamma $
is successor or if
$ \gamma $
is limit and we want
$ \gamma \in \mathrm {Ord}_{\mathrm {SD}}(X) $
, while we set
$ X = (\beta \times K_\alpha ) \oplus Z_\gamma $
if
$ \gamma $
is limit and we want
$ \gamma \in \mathrm {Ord}_{\mathrm {NSD}}(X) $
.
(2) Consider the space
$ \hat {\mathcal {C}} = {}^\omega \{ 0,2 \} \subseteq \mathcal {N} $
, which is trivially homeomorphic to
$ \mathcal {C} $
, and recall the spaces
$ Y_\delta $
and
$ Z_\delta $
from part (1). If
$ \gamma < \alpha $
, then we first buildFootnote
13
$ \hat {\mathcal {C}}_\alpha = \hat {\mathcal {C}} \cup K_{\alpha +1} $
, so that
$ ||\hat {\mathcal {C}}_\alpha ||_{\operatorname {CB}} = \alpha $
and
$ \operatorname {ker}_{\operatorname {CB}}(\hat {\mathcal {C}}_\alpha ) = \hat {\mathcal {C}} $
is compact, and then set
$ X = \hat {\mathcal {C}}_\alpha \oplus Y_\gamma $
or
$ X = \hat {\mathcal {C}}_\alpha \oplus Z_\gamma $
depending on whether we want
$ \gamma \in \mathrm {Ord}_{\mathrm {SD}}(X) $
or not in case
$ \gamma $
is limit. If instead
$ \gamma = \alpha $
, then we set
$ X = \hat {\mathcal {C}} \oplus Y_\gamma $
or
$ X = \hat {\mathcal {C}} \cup Z_\gamma $
, again depending on the given requirement on selfduality of
$\gamma $
when it is limit. (Notice that
$ ||\hat {\mathcal {C}} \cup Z_\gamma ||_{\operatorname {CB}} = \gamma $
because
$ (0)^{\infty } $
is no longer isolated in
$ D^{\gamma }_{\operatorname {CB}}(\hat {\mathcal {C}} \cup Z_\gamma ) = \hat {\mathcal {C}} $
.)
(3) Set
$ X = \mathcal {N} \oplus \hat {\mathcal {C}}_\alpha $
, where
$ \hat {\mathcal {C}}_\alpha $
is as in part (2).
5 Proof of Main Theorem 2
The Wadge hierarchy
over a (zero-dimensional) Polish space provides the finest possible measurement of the topological complexity of its subsets. Slightly changing our perspective, it is natural to ask whether it is possible to classify zero-dimensional Polish spaces up to the measurement scale
they give rise to. More formally, consider the standard Effros–Borel space
$ F(\mathcal {N}) $
of all closed subsets of
$ \mathcal {N} $
, which up to homeomorphism contains all zero-dimensional Polish spaces, and for
$ X,Y \in F(\mathcal {N}) $
set
Remark 5.1. The current formulation of the problem is within our axiomatic setup
$ \mathsf {ZF} + \mathsf {DC}(\mathbb {R}) + \mathsf {AD} $
. If one wants to dispense with any determinacy assumption, the problem could be formulated by requiring that
and
be isomorphic when restricted to Borel sets, or even just to sets in
$ \boldsymbol {\Delta }^0_2 $
: all results we are going to present would be unaffected by such a change, thus providing determinacy-free anti-classification results.
The relation
$ \approx _{\mathsf {W}} $
is strictly coarser than closed biembeddability: by Lemma 3.1, if
$ X $
and
$ Y $
are such that there are closed embeddings from each space into the other one, then
$ X \approx _{\mathsf {W}} Y $
; but on the other hand
$ \mathcal {N} \approx _{\mathsf {W}} \omega \times \mathcal {C} $
even though
$ \mathcal {N} $
is not homeomorphic to a closed subset of
$ \omega \times \mathcal {C} $
.
Our analysis provides a quite satisfactory solution to the classification problem associated with
$ \approx _{\mathsf {W}} $
by providing a natural assignment of complete invariants. More precisely, call a pair of ordinals
$ (\alpha ,\beta ) $
a Wadge invariant if it satisfies the following conditions:
-
•
$ \alpha ,\beta \in \omega _1 \cup \{ \Theta \} $
and
$ \alpha \leq \beta $
. -
• If
$ \beta \neq \Theta $
, then
$ \beta $
is at least
$ 2 $
and is a successor ordinal. -
• If
$ \alpha \neq \Theta $
, then either
$ \alpha = 0 $
, or
$ \alpha = \lambda +1 $
for some countable limit ordinal
$ \lambda $
, or else
$ \alpha = \lambda $
for some countable ordinal
$ \lambda $
which is a limit of limit ordinals.
Notice that each Wadge invariant can easily be coded into a single countable ordinal. For each zero-dimensional Polish space
$ X $
, the pair
$ (\alpha _X, \Theta _X ) $
is a Wadge invariant which completely determines the structure of
up to order-isomorphism (by the alternating property, Proposition 4.7, and Corollary 4.14). This means that the map
$ X \mapsto (\alpha _X, \Theta _X) $
is an assignment of complete invariants for the classification of zero-dimensional Polish spaces up to
$ \approx _{\mathsf {W}} $
.
Remark 5.2. As briefly discussed in Footnote 12 on page 21, in the
$ \mathsf {AD} $
-world we must make sure that when
$ (\alpha _X,\Theta _X) = (\alpha _Y,\Theta _Y) $
then we can actually find an isomorphism between
and
without appealing to the Axiom of Choice
$ \mathsf {AC} $
. But this is granted by the fact that by Theorem 2.9(2), Corollary 3.19, and Lemma 3.4, if
$ X \subseteq \mathcal {N} $
is closed and
$ \rho \colon \mathcal {N} \to X $
is a retraction then the map
$ \iota _X $
defined by
$ \boldsymbol {\Gamma } \mapsto \rho ^{-1}(\boldsymbol {\Gamma }) $
is an embedding of
$ \langle \mathrm {Class}_{\mathsf {W}}(X), { \subseteq } \rangle $
into
$ \langle \mathrm {Class}_{\mathsf {W}}(\mathcal {N}), { \subseteq } \rangle $
sending
$ \mathrm {Class}_{\operatorname {NSD}}(X) $
into a suitable initial segment of
$ \mathrm {Class}_{\operatorname {NSD}}(\mathcal {N}) $
. By our analysis, it then turns out that
$ \iota _Y^{-1} \circ \iota _X $
is the required canonical isomorphism between
$ \langle \mathrm {Class}_{\mathsf {W}}(X), { \subseteq } \rangle $
and
$ \langle \mathrm {Class}_{\mathsf {W}}(Y), { \subseteq } \rangle $
.
Conversely, by Proposition 4.16 together with the computation of
$ \alpha _X $
provided after Corollary 4.14 and the computation of the length of
from (the proof of) Theorem 4.15, we get a zero-dimensional Polish space
$ X $
with
$ (\alpha _X, \Theta _X) = (\alpha ,\beta ) $
for every Wadge invariant
$ (\alpha ,\beta ) $
. In particular, the
$ \approx _{\mathsf {W}} $
-equivalence classes can be well-ordered in order type
$ \omega _1 $
.
In contrast to these “positive” classification results, we are now going to show that, except for the case when
$ X $
is a singleton, it is difficult to determine whether
$ X $
belongs to a given
$ \approx _{\mathsf {W}} $
-equivalence class: given a Wadge invariant
$ (\alpha ,\beta ) \neq (0,2) $
, the set of zero-dimensional Polish spaces
$ X $
with
$ (\alpha _X , \Theta _X) = (\alpha ,\beta ) $
is never Borel. This proves Main Theorem 2.
Theorem 5.3 (
$ \mathsf {AD} $
).
-
(1) The class
$ [\mathcal {N}]_{\approx _{\mathsf {W}}} $
corresponding to the Wadge invariant
$ (\Theta ,\Theta ) $
is a Borel complete
$ \boldsymbol {\Sigma }^1_1 $
set. -
(2) For any
$ X \in F(\mathcal {N}) $
with
$ |X|> 1 $
and
$ X \not \approx _{\mathsf {W}} \mathcal {N} $
, the set
$ [X]_{\approx _{\mathsf {W}}} $
is Borel
$ \boldsymbol {\Pi }^1_1 $
-hard. -
(3) For every countableFootnote 14 limit ordinal
$\alpha $
, the set
$ \{ X \in F(\mathcal {N}) \mid \alpha \in \mathrm {Ord}_{\mathrm {SD}}(X) \} $
is Borel
$ \boldsymbol {\Sigma }^1_1 $
-hard.
As the proof will show, the same is true if we restrict the ambient space to the analytic space
$ F_{> \aleph _0}(\mathcal {N}) $
of uncountable zero-dimensional Polish spaces, i.e., to spaces with Wadge invariant of the form
$ (\alpha ,\Theta ) $
. (Just replace the Borel map
$ F $
with
$ F_{\mathcal {C}} $
everywhere.)
Proof We use the following variation of a construction from [Reference Camerlo, Marcone and Motto Ros7]. Fix a bijection
$ \langle \cdot , \cdot \rangle \colon \omega ^2 \to \omega \setminus \{ 0 \} $
. For
$ s \in {}^{<\omega }(\omega \setminus \{ 0 \}) $
let
$ \pi (s) = t $
if and only if
$ \operatorname {lh}(t) = \operatorname {lh}(s) $
and for all
$ i < \operatorname {lh}(s) $
there is
$ n_i \in \omega $
such that
$ s(i) = \langle t(i), n_i \rangle $
. Define the Borel map
$ F \colon \mathrm {Tr} \to F(\mathcal {N}) $
from the Polish space
$ \mathrm {Tr} $
of trees on
$\omega $
into the space of (codes for) zero-dimensional Polish spaces by letting
$ F(T) $
be the body of the pruned tree
$$\begin{align*}\{ s {^{\smallfrown}} (0)^k \mid s \in {}^{< \omega}(\omega \setminus \{ 0 \} ) \wedge \pi(s) \in T \wedge k \in \omega \}. \end{align*}$$
If
$ x \in \mathcal {N} $
is an infinite branch through
$ T $
, then
$ F(T) $
contains a closed set homeomorphic to
$ \mathcal {N} $
, namely, the set of all
$ y \in \mathcal {N} $
such that
$ y \restriction n \in {}^{< \omega }(\omega \setminus \{ 0 \}) $
and
$ \pi (y \restriction n) = x \restriction n $
for all
$ n \in \omega $
. Hence
$ F(T) $
is not
$\sigma $
-compact and
$ \operatorname {ker}_{\operatorname {CB}}(F(T)) $
is not compact. Conversely, if
$ T $
is well-founded then all elements of
$ F(T) $
are of the form
$ s {^{\smallfrown }} (0)^\infty $
for some
$ s \in {}^{< \omega }(\omega \setminus \{ 0 \}) $
and they are isolated in
$ F(T) $
, i.e.,
$ F(T)$
is discrete.
To prove (1), first recall that by Corollary 4.10 we have
$ X \approx _{\mathsf {W}} \mathcal {N} $
if and only if
$ \operatorname {ker}_{\operatorname {CB}}(X) $
is not compact. But
$ \operatorname {ker}_{\operatorname {CB}}(X) $
is not compact if and only if there is
$ H \in F(\mathcal {N}) $
such that
$ H \subseteq X $
,
$ H $
is perfect, and
$ H \notin K(\mathcal {N}) $
, where
$ K(\mathcal {N}) $
is the collection of all compact subspaces of
$ \mathcal {N} $
. Since
$ K(\mathcal {N}) $
is Borel in
$ F (\mathcal {N}) $
and “
$ H $
is perfect” is a Borel statement (see the proof of [Reference Kechris17, Theorem 27.5]), this is a
$ \boldsymbol {\Sigma }^1_1 $
definition of
$ [\mathcal {N}]_{\approx _{\mathsf {W}}} $
. For the hardness part, it is enough to use the map
$ F $
from the previous paragraph, recalling that the set of ill-founded trees is a complete
$ \boldsymbol {\Sigma }^1_1 $
set.
Consider now a space
$ X $
as in (2), and let
$ F_X \colon \mathrm {Tr} \to F(\mathcal {N}) $
be the Borel map defined by
$ F_X(T) = F(T) \oplus X $
, where
$ F $
is as in the first paragraph. If
$ T $
is ill-founded then
$ F_X(T) \approx _{\mathsf {W}} \mathcal {N} $
because
$ \operatorname {ker}_{\operatorname {CB}}(F_X(T)) \supseteq \operatorname {ker}_{\operatorname {CB}}(F(T)) $
is not compact. If instead
$ T $
is well-founded, then
$ F(T) $
is a discrete clopen subset of
$ F_X(T) $
. Then
$ ||F_X(T)||_{\operatorname {CB}} = ||X||_{\operatorname {CB}} $
, and (when this makes sense)
$ F_X(T) $
is simple if and only if so is
$ X $
because we assumed
$ |X|> 1 $
. Finally,
$ ||F_X(T)||_{\operatorname {Comp}} = \max \{ 1, ||X||_{\operatorname {Comp}} \} $
, hence
$ \alpha _{F_X(T)} = \alpha _X $
. It follows that
$ F_X(T) \approx _{\mathsf {W}} X $
. This shows that
$ F_X $
reduces the set of well-founded trees to
$ [X]_{\approx _{\mathsf {W}}} $
, hence the latter is Borel
$ \boldsymbol {\Pi }^1_1 $
-hard.
Part (3) follows from part (2); for example, one can use the map
$ F_{\mathcal {C}} $
.
Remark 5.4. Theorem 5.3 admits several variations. For example, for all relevant
$ \alpha $
the set
$ \{ X \in F(\mathcal {N}) \mid \alpha _X = \alpha \} $
is either Borel
$ \boldsymbol {\Sigma }^1_1 $
-hard (if
$ \alpha = \Theta $
) or Borel
$ \boldsymbol {\Pi }^1_1 $
-hard (if
$ \alpha $
is countable). Similarly, for all relevant
$\beta $
the set
$ \{ X \in F(\mathcal {N}) \mid \Theta _X = \beta \} $
is either Borel
$ \boldsymbol {\Sigma }^1_1 $
-hard (if
$ \beta = \Theta $
) or Borel
$ \boldsymbol {\Pi }^1_1 $
-hard (if
$ \beta < \omega _1 $
).
Theorem 5.3 implies that the equivalence relation
$ \approx _{\mathsf {W}} $
is neither analytic nor co-analytic. Therefore, although we obtained a quite simple and satisfactory classification in terms of Wadge invariants (which are, essentially, countable ordinals), there is no reasonable classification for
$ \approx _{\mathsf {W}} $
in terms of, say, Borel reducibility. In a sense, this hints at the fact that there are natural situations in which the latter is not an accurate tool to determine the complexity of the classification problem at hand.
The situation radically changes if we instead consider only perfect zero-dimensional Polish spaces.
Proposition 5.5 (
$ \mathsf {AD} $
).
Let
$ F_{\mathrm {p}}(\mathcal {N}) $
be the standard Borel space of perfect zero-dimensional Polish spaces, and let
$ X \in F_{\mathrm {p}}(\mathcal {N}) $
. Then either
$ X \approx _{\mathsf {W}} \mathcal {N} $
or
$ X \approx _{\mathsf {W}} \mathcal {C} $
, and those two
$ \approx _{\mathsf {W}} $
-equivalence classes are Borel.
Proof All spaces
$ X \in F_{\mathrm {p}}(\mathcal {N}) $
are uncountable, and thus
$ \Theta _X = \Theta $
by Proposition 2.5. Moreover, we have
$ \operatorname {ker}_{\operatorname {CB}}(X) = X $
, thus
$ X \approx _{\mathsf {W}} \mathcal {N} $
if
$ X $
is not compact (Corollary 4.10), while
$ X \approx _{\mathsf {W}} \mathcal {C} $
if
$ X $
is compact (Proposition 3.2(2)). Finally, the distinction between the two cases is Borel because
$ K(\mathcal {N}) $
is Borel in
$ F(\mathcal {N}) $
.
For the record, we also notice that the proof of Theorem 5.3(1) actually gives a seemingly unrelated complexity result that might be of independent interest.
Proposition 5.6. The set
$ \{ X \in F(\mathcal {N}) \mid \operatorname {ker}_{\operatorname {CB}}(X) \text { is compact } \} $
is a Borel complete
$ \boldsymbol {\Pi }^1_1 $
set.
6 Some questions
Our analysis raises a number of natural questions: we just mention a few of them which follow closely our approach.
With the exception of
$ [\mathcal {N}]_{\approx _{\mathsf {W}}} $
, Theorem 5.3 only gives lower bounds for the complexity of the other
$ \approx _{\mathsf {W}} $
-equivalence classes, but they might not be sharp.
Question 6.1. Let
$ X $
be a zero-dimensional Polish space with at least two points and
$ \operatorname {ker}_{\operatorname {CB}}(X) $
not compact. What is the exact complexity of
$ [X]_{\approx _{\mathsf {W}}} $
? What for
$ X = \mathcal {C} $
? What is the complexity of the equivalence relation
$ \approx _{\mathsf {W}} $
as a subset of the square
$ F(\mathcal {N}) \times F(\mathcal {N}) $
?
Similar questions can be raised for Polish spaces of positive dimension, or for the natural adaptation of
$ \approx _{\mathsf {W}} $
to the hyperspace
$ F({}^\omega \mathbb {R}) $
of all Polish spaces.
Our results are limited to zero-dimensional Polish spaces, and it is natural to wonder what happens for other kinds of Polish spaces. As recalled in the introduction, this line of research has been pursued in many papers and by various authors, although we are still lacking a complete description of the various possibilities. But from a different perspective, the restriction to zero-dimensional Polish spaces is equivalent to considering only closed (or even just
$ G_\delta $
) subspaces of
$ \mathcal {N} $
. One can then wonder what happens for arbitrary subspaces of
$ \mathcal {N} $
, starting with the Borel ones. This is a quite different problem, as we retain zero-dimensionality (together with separability and metrizability) but drop the completeness of the space. We leave this research direction open for future investigations.
Acknowledgment
The authors are grateful to the anonymous referee for their useful suggestions and references.
Funding
This work was supported by the project PRIN 2017 “Mathematical Logic: models, sets, computability” (prot. 2017NWTM8R) and by the “National Group for Algebraic and Geometric Structures, and their Applications” (GNSAGA–INDAM).
