Compact Spaces Via \(p\)-closed Subsets

Abstract

In [2] the following problems were listed as open: Problem 14 [1]. Is a regular space in which every closed subset is regular-closed compact? Problem 15. Is a Urysohn-space in which every closed subset is Urysohnclosed compact? To answer the question for Hausdorff-closed spaces in the affirmative, M. H. Stone [18] used Boolean rings and M. Katˇetov [6] used topological methods. In [15], a different method was used to answer the question for Hausdorff-closed spaces and in [16], the other two questions were answered in
the affirmative. In this paper, different proofs from those in [15] and [16] are given answering all of these questions. An improvement is given as follows. If every closed subset of a Hausdorff (Urysohn)[regular] space \(X\) is an H-set (a U-set)[an R-set] then \(X\) is compact. An affirmative answer is also given to a question posed by Girou [7] and Vermeer [20] as an open question: Is a Hausdorff-closed space in which all of its Hausdorff-closed subspaces are minimal Hausdorff compact? Similar questions are also answered in the affirmative for Urysohn-closed and regular-closed spaces.