Note on separate continuity and the Namioka property
by
Dennis Burke
Miami University
Coauthors: Roman Pol (Miami University and Warsaw University)

This is joint work with Roman Pol. All spaces are completely regular.

I. Namioka has shown that for any separately continuous f:B×K --> R, with B Cech-complete and K compact, there is a dense A subset or equal B such that f is (jointly) continuous on A×K.

Haydon has shown that the conclusion of the above can fail for B a Choquet space and K compact scattered. We offer another example:

Theorem. There is a Choquet space B and separately continuous f:B×\betaB --> R such that the restriction f|\Delta to the diagonal does not have a dense set of continuity points.

However, for K a compact fragmentable space we have:

Theorem. For any separately continuous f:T×K --> R, where K is a compact fragmentable space, and for any Baire subspace F of T×K the set of points of continuity of f|F:F --> R is dense in F.

In particular, if (in the above theorem) F=T×K is Baire then f would have a dense set of points of continuity, a result of Kenderov, Kortezov and Moors.

We say that <B, K> is a weak-Namioka pair if K is compact and for any separately continuous f:B×K --> R and a closed subset F projecting irreducibly onto B, the set of points of continuity of f|F is dense in F.

The first theorem shows that some pairs <B, \betaB>, with B Baire, may not be weak-Namioka pairs. The second theorem shows that each <B, K>, with B Baire and K compact fragmentable is a weak-Namioka pair.

Proposition. If, for every compact K, the pair <T, K> is a weak-Namioka pair then T is a Baire space.

Date received: February 22, 2002