Let Γ be a class of subsets of Baire space (ωω) closed under inverse images by continuous functions. We say such a Γ is continuously closed. Let
, the class dual to Γ, consist of the complements relative to ωω of members of Γ. If Γ is not selfdual, i.e.,
, then let
. A continuously closed nonselfdual class Γ of subsets of ωΓ is said to have the first separation property [2] if
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0022481200049902/resource/name/S0022481200049902_eqnU1.gif?pub-status=live)
The set C is said to separate A and B. The class Γ is said to have the second separation property [3] if
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0022481200049902/resource/name/S0022481200049902_eqnU2.gif?pub-status=live)
We shall assume the axiom of determinateness and show that if Γ is a continuously closed class of subsets of ωω and
then
(1) Γ has the first separation property iff
does not have the second separation property, and
(2) either Γ or
has the second separation property.
Of course, (1) and (2) taken together imply that Γ and
cannot both have the first separation property.