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Countable unions of compact spaces with the Namioka property

Part of: Basic constructions

Published online by Cambridge University Press:  26 February 2010

Richard Haydon
Richard Haydon
Affiliation:
Brasenose College, Oxford, OX1 4AJ
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A compact space K is said to have the Namioka Property, or to belong to the class *, if, for every Baire space B and every separately continuous function Ψ:B × K → ℝ, there is dense δ subset H of B such that Ψ is (jointly) continuous at all points of H × K. Although the terminology is more recent, the idea of looking at properties of this kind goes back to Namioka's paper [6] on separate and joint continuity. Talagrand [8] gave the first example of a compact space that is not in * and it is now known [4] that there are even examples of scattered compact spaces that are not in *. On the other hand, many good classes of compact spaces have been shown to be contained in *, probably the most general being the class of continuous images of Valdivia compacts [2]. The aim of this note is to prove the following stability result: a compact space which is a countable union of closed subsets with the Namioka Property does itself possess that property.

MSC classification

Secondary: 54B10: Product spaces
Type
Research Article
Information
Mathematika , Volume 41 , Issue 1 , June 1994 , pp. 141 - 144
Copyright
Copyright © University College London 1994

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References

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