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A characterization of Corson-compact spaces. (English) Zbl 0769.54025
A space $$X$$ is Corson-compact if it is a compact subset of a $$\Sigma$$- product of the reals (i.e. those members of $${\mathfrak R}^ I$$ which are 0 on all but countably many coordinates). There are several characterizations of Corson-compact spaces, for example, as those spaces which possess a point-countable separating collection of open $$F_ \sigma$$-sets. In the present article, the author obtains a characterization in terms of countable elementary substructures (in the sense of set-theory). In the reviewer’s terms, if $$M$$ is an elementary submodel of a sufficient large fragment of the universe which contains a Corson-compact space $$X$$ (together with its topology $$T$$), then the collection of sets $$T\cap M$$ is large enough to induce the subspace topology on the closure (in $$X$$) of $$X\cap M$$. The necessity of this statement is quite straightforward, but the proof of the sufficiency is more involved. A nice application of the characterization is a very simple proof of the fact that a continuous image of a Corson-compact is again a Corson-compact.
Reviewer: A.Dow (North York)

##### MSC:
 54D30 Compactness
##### Keywords:
elementary submodel; Corson-compact space
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