A quasi-closure preserving sum theorem about the Namioka property
✍ Scribed by Ahmed Bouziad
- Book ID
- 104295212
- Publisher
- Elsevier Science
- Year
- 1997
- Tongue
- English
- Weight
- 535 KB
- Volume
- 81
- Category
- Article
- ISSN
- 0166-8641
No coin nor oath required. For personal study only.
✦ Synopsis
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 f : B x X ----t Ii3 there exists a G6 dense subset A of B such that f is (jointly) continuous at each point of A x X. A collection A of subsets of a topological space X is said to be quasi-closure preserving if all countable subcollections of A are closure preserving.
Let X be a compact space. The principal result of this note is slightly more genera1 than the following statement: If there exists a quasi-closure preserving collection A of co-Namioka compact subspaces of X such that X = U A, 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.