A characterization of Corson-compact spaces.

*(English)*Zbl 0769.54025A 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 |