[EN] Every first countable pseudocompact Tychonoff space X has the property that every pseudocompact subspace of X is a closed subset of X (denoted herein by “FCC”). We study the property FCC and several closely related ones, and focus on the behavior of extension and other spaces which have one or more of these properties. Characterization, embedding and product theorems are obtained, and some examples are given which provide results such as the following. There exists a separable Moore space which has no regular, FCC extension space. There exists a compact Hausdorff Fréchet space which is not FCC. There exists a compact Hausdorff Fréchet space X such that X, but not X2, is FCC.

The first author gratefully acknowledges partial research support from the National Science Foundation, Grant No. 2975010131. The third and fourth authors gratefully acknowledge partial research support from the University of Kansas and the sabbatical leave programs of their respective institutions, and in the case of the fourth author, from the Natural Sciences and Engineering Research Council of Canada.