A quasi-closure preserving sum theorem about the Namioka property
Résumé
A compact space X is said to be co-Namioka (or to have the Namioka property) if, for every Baire space B and every separately continuous function ƒ: B × X → R there exists a $G_δ$ dense subset A of B such that ƒ is (jointly) continuous at each point of A × X. A collection $\cal A$ of subsets of a topological space X is said to be quasi-closure preserving if all countable subcollections of $\cal A are closure preserving. Let X be a compact space. The principal result of this note is slightly more general than the following statement: If there exists a quasi-closure preserving collection $\cal A$ of co-Namioka compact subspaces of X the union of whic is dense in X, then X is co-Namioka. As an application of this property, we show that the Alexandroff compactification of every locally compact scattered space, which is hereditarily submetacompact, is co-Namioka. In particular, every compact scattered hereditarily submetacompact space has the Namioka property.