We examine a semigroup analogue of the Kumjian–Renault representation of C*-algebras with Cartan subalgebras on twisted groupoids. Specifically, we represent semigroups with distinguished normal subsemigroups as ‘slice-sections’ of groupoid bundles.

We introduce natural strengthenings of sequential compactness, the r-Ramsey property for each natural number $r\geq 1$ . We prove that metrizable compact spaces are r-Ramsey for all r and give examples of compact spaces that are r-Ramsey but not $(r+1)$ -Ramsey for each $r\geq 1$ (assuming Continuum Hypothesis (CH) for all $r>1$ ). Productivity of the r-Ramsey property is considered.

A $k_{\omega }$ -space X is a Hausdorff quotient of a locally compact, $\sigma $ -compact Hausdorff space. A theorem of Morita’s describes the structure of X when the quotient map is closed, but in 2010 a question of Arkhangel’skii’s highlighted the lack of a corresponding theorem for nonclosed quotient maps (even from subsets of $\mathbb {R}^n$ ). Arkhangel’skii’s specific question had in fact been answered by Siwiec in 1976, but a general structure theorem for $k_{\omega }$ -spaces is still lacking. We introduce pure quotient maps, extend Morita’s theorem to these, and use Fell’s topology to show that every quotient map can be “purified” (and thus every $k_{\omega }$ -space is the image of a pure quotient map). This clarifies the structure of arbitrary $k_{\omega }$ -spaces and gives a fuller answer to Arkhangel’skii’s question.

Let G be a locally compact group and let ${\mathcal {SUB}(G)}$ be the hyperspace of closed subgroups of G endowed with the Chabauty topology. The main purpose of this paper is to characterise the connectedness of the Chabauty space ${\mathcal {SUB}(G)}$ . More precisely, we show that if G is a connected pronilpotent group, then ${\mathcal {SUB}(G)}$ is connected if and only if G contains a closed subgroup topologically isomorphic to ${{\mathbb R}}$ .

We prove that the modal logic of a crowded locally compact generalized ordered space is $\textsf {S4}$ . This provides a version of the McKinsey–Tarski theorem for generalized ordered spaces. We then utilize this theorem to axiomatize the modal logic of an arbitrary locally compact generalized ordered space.

We continue the research of the relation $\hspace {1mm}\widetilde {\mid }\hspace {1mm}$ on the set $\beta \mathbb {N}$ of ultrafilters on $\mathbb {N}$ , defined as an extension of the divisibility relation. It is a quasiorder, so we see it as an order on the set of $=_{\sim }$ -equivalence classes, where $\mathcal {F}=_{\sim }\mathcal {G}$ means that $\mathcal {F}$ and $\mathcal {G}$ are mutually $\hspace {1mm}\widetilde {\mid }$ -divisible. Here we introduce a new tool: a relation of congruence modulo an ultrafilter. We first recall the congruence of ultrafilters modulo an integer and show that $=_{\sim }$ -equivalent ultrafilters do not necessarily have the same residue modulo $m\in \mathbb {N}$ . Then we generalize this relation to congruence modulo an ultrafilter in a natural way. After that, using iterated nonstandard extensions, we introduce a stronger relation, which has nicer properties with respect to addition and multiplication of ultrafilters. Finally, we introduce a strengthening of $\hspace {1mm}\widetilde {\mid }\hspace {1mm}$ and show that it also behaves well with respect to the congruence relation.

We prove that the existence of a measurable cardinal is equivalent to the existence of a normal space whose modal logic coincides with the modal logic of the Kripke frame isomorphic to the powerset of a two element set.

We investigate C-sets in almost zero-dimensional spaces, showing that closed $\sigma $ C-sets are C-sets. As corollaries, we prove that every rim- $\sigma $ -compact almost zero-dimensional space is zero-dimensional and that each cohesive almost zero-dimensional space is nowhere rational. To show that these results are sharp, we construct a rim-discrete connected set with an explosion point. We also show that every cohesive almost zero-dimensional subspace of $($ Cantor set $)\!\times \mathbb R$ is nowhere dense.

Urysohn’s lemma is a crucial property of normal spaces that deals with separation of closed sets by continuous functions. It is also a fundamental ingredient in proving the Tietze extension theorem, another property of normal spaces that deals with the existence of extensions of continuous functions. Using the Cantor function, we give alternative proofs for Urysohn’s lemma and the Tietze extension theorem.

We establish that the existence of a winning strategy in certain topological games, closely related to a strong game of Choquet, played in a topological space $X$ and its hyperspace $K(X)$ of all nonempty compact subsets of $X$ equipped with the Vietoris topology, is equivalent for one of the players. For a separable metrizable space $X$, we identify a game-theoretic condition equivalent to $K(X)$ being hereditarily Baire. It implies quite easily a recent result of Gartside, Medini and Zdomskyy that characterizes hereditary Baire property of hyperspaces $K(X)$ over separable metrizable spaces $X$ via the Menger property of the remainder of a compactification of $X$. Subsequently, we use topological games to study hereditary Baire property in spaces of probability measures and in hyperspaces over filters on natural numbers. To this end, we introduce a notion of strong $P$-filter ${\mathcal{F}}$ and prove that it is equivalent to $K({\mathcal{F}})$ being hereditarily Baire. We also show that if $X$ is separable metrizable and $K(X)$ is hereditarily Baire, then the space $P_{r}(X)$ of Borel probability Radon measures on $X$ is hereditarily Baire too. It follows that there exists (in ZFC) a separable metrizable space $X$, which is not completely metrizable with $P_{r}(X)$ hereditarily Baire. As far as we know, this is the first example of this kind.

We show in ZFC that the existence of completely separable maximal almost disjoint families of subsets of $\omega $ implies that the modal logic $\mathbf {S4.1.2}$ is complete with respect to the Čech–Stone compactification of the natural numbers, the space $\beta \omega $ . In the same fashion we prove that the modal logic $\mathbf {S4}$ is complete with respect to the space $\omega ^*=\beta \omega \setminus \omega $ . This improves the results of G. Bezhanishvili and J. Harding in [4], where the authors prove these theorems under stronger assumptions ( $\mathfrak {a=c}$ ). Our proof is also somewhat simpler.

To study when a paratopological group becomes a topological group, Arhangel’skii et al. [‘Topological games and topologies on groups’, Math. Maced. 8 (2010), 1–19] introduced the class of $(\,\unicode[STIX]{x1D6FD},G_{\unicode[STIX]{x1D6F1}})$-unfavourable spaces. We show that every $\unicode[STIX]{x1D707}$-complete (or normal) $(\,\unicode[STIX]{x1D6FD},G_{\unicode[STIX]{x1D6F1}})$-unfavourable semitopological group is a topological group. We prove that the product of a $(\,\unicode[STIX]{x1D6FD},G_{\unicode[STIX]{x1D6F1}})$-unfavourable space and a strongly Fréchet $(\unicode[STIX]{x1D6FC},G_{\unicode[STIX]{x1D6F1}})$-favourable space is $(\,\unicode[STIX]{x1D6FD},G_{\unicode[STIX]{x1D6F1}})$-unfavourable. We also show that continuous closed irreducible mappings preserve the $(\,\unicode[STIX]{x1D6FD},G_{\unicode[STIX]{x1D6F1}})$-unfavourableness in both directions.

We present a result about $G_{\unicode[STIX]{x1D6FF}}$ covers of a Hausdorff space that implies various known cardinal inequalities, including the following two fundamental results in the theory of cardinal invariants in topology: $|X|\leqslant 2^{L(X)\unicode[STIX]{x1D712}(X)}$ (Arhangel’skiĭ) and $|X|\leqslant 2^{c(X)\unicode[STIX]{x1D712}(X)}$ (Hajnal–Juhász). This solves a question that goes back to Bell, Ginsburg and Woods’s 1978 paper (M. Bell, J.N. Ginsburg and R.G. Woods, Cardinal inequalities for topological spaces involving the weak Lindelöf number, Pacific J. Math. 79(1978), 37–45) and is mentioned in Hodel’s survey on Arhangel’skiĭ’s Theorem (R. Hodel, Arhangel’skii’s solution to Alexandroff’s problem: A survey, Topology Appl. 153(2006), 2199–2217).

In contrast to previous attempts, we do not need any separation axiom beyond $T_{2}$.

We study the following problem: For which Tychonoff spaces $X$ do the free topological group $F(X)$ and the free abelian topological group $A(X)$ admit a quotient homomorphism onto a separable and nontrivial (i.e., not finitely generated) group? The existence of the required quotient homomorphisms is established for several important classes of spaces $X$, which include the class of pseudocompact spaces, the class of locally compact spaces, the class of $\unicode[STIX]{x1D70E}$-compact spaces, the class of connected locally connected spaces, and some others.

We also show that there exists an infinite separable precompact topological abelian group $G$ such that every quotient of $G$ is either the one-point group or contains a dense non-separable subgroup and, hence, does not have a countable network.

It is proved that the free topological vector space $\mathbb{V}([0,1])$ contains an isomorphic copy of the free topological vector space $\mathbb{V}([0,1]^{n})$ for every finite-dimensional cube $[0,1]^{n}$, thereby answering an open question in the literature. We show that this result cannot be extended from the closed unit interval $[0,1]$ to general metrisable spaces. Indeed, we prove that the free topological vector space $\mathbb{V}(X)$ does not even have a vector subspace isomorphic as a topological vector space to $\mathbb{V}(X\oplus X)$, where $X$ is a Cook continuum, which is a one-dimensional compact metric space. This is also shown to be the case for a rigid Bernstein set, which is a zero-dimensional subspace of the real line.

It is proved that the Continuum Hypothesis implies that any sequence of rapid P-points of length ${<}\mathfrak{c}^{+}$ that is increasing with respect to the Rudin–Keisler ordering is bounded above by a rapid P-point. This is an improvement of a result from B. Kuzeljevic and D. Raghavan. It is also proved that Jensen’s diamond principle implies the existence of an unbounded strictly increasing sequence of P-points of length $\unicode[STIX]{x1D714}_{1}$ in the Rudin–Keisler ordering. This shows that restricting to the class of rapid P-points is essential for the first result.

The notion of metric compactification was introduced by Gromov and later rediscovered by Rieffel. It has been mainly studied on proper geodesic metric spaces. We present here a generalization of the metric compactification that can be applied to infinite-dimensional Banach spaces. Thereafter we give a complete description of the metric compactification of infinite-dimensional $\ell _{p}$ spaces for all $1\leqslant p<\infty$. We also give a full characterization of the metric compactification of infinite-dimensional Hilbert spaces.

For a Tychonoff space $X$, let $\mathbb{V}(X)$ be the free topological vector space over $X$, $A(X)$ the free abelian topological group over $X$ and $\mathbb{I}$ the unit interval with its usual topology. It is proved here that if $X$ is a subspace of $\mathbb{I}$, then the following are equivalent: $\mathbb{V}(X)$ can be embedded in $\mathbb{V}(\mathbb{I})$ as a topological vector subspace; $A(X)$ can be embedded in $A(\mathbb{I})$ as a topological subgroup; $X$ is locally compact.

We introduce the open degree of a compact space, and we show that for every natural number $n$, the separable Rosenthal compact spaces of degree $n$ have a finite basis.

Let A = C(X) ⊗ K(H), where X is a compact Hausdorff space and K(H) is the algebra of compact operators on a separable infinite-dimensional Hilbert space. Let As be the algebra of strong*-continuous functions from X to K(H). Then As/A is the inner corona algebra of A. We show that if X has no isolated points, then As/A is an essential ideal of the corona algebra of A, and Prim(As/A), the primitive ideal space of As/A, is not weakly Lindelof. If X is also first countable, then there is a natural injection from the power set of X to the lattice of closed ideals of As/A. If X = βℕ\ℕ and the continuum hypothesis (CH) is assumed, then the corona algebra of A is a proper subalgebra of the multiplier algebra of As/A. Several of the results are obtained in the more general setting of C0(X)-algebras.