Joint continuity of separately continuous functions

Abstract

It is shown that a separately continuous function f : X × Y → Z f:X \times Y \to Z from the product of a certain type of Hausdorff space X X and a compact Hausdorff space Y Y into a metrizable space Z Z is jointly continuous on a set of the type A × Y A \times Y , where A A is a dense G δ {G_\delta } set in X X . The class of Hausdorff spaces X X in question is defined by a gametheoretic condition. The result improves (and simplifies the proof of) a recent result of Namioka. Many "deep" theorems in functional analysis and automatic continuity theory are easy corollaries.