Note on separate continuity and the Namioka property

Abstract

A pair 〈 B , K 〉 is a Namioka pair if K is compact and for any separately continuous f : B × K → R , there is a dense A ⊆ B such that f is ( jointly) continuous on A × K . We give an example of a Choquet space B and separately continuous f : B × β B → R such that the restriction f | Δ to the diagonal does not have a dense set of continuity points. However, for K a compact fragmentable space we have: For any separately continuous f : T × K → R and for any Baire subspace F of T × K , the set of points of continuity of f | F : F → R is dense in F. 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. We show that T is a Baire space if the pair 〈 T , K 〉 is a weak-Namioka pair for every compact K. Under (CH) there is an example of a space B such that 〈 B , K 〉 is a Namioka pair for every compact K but there is a countably compact C and a separately continuous f : B × C → R which has no dense set of continuity points; in fact, f does not even have the Baire property.