Compact Spaces Via \(p\)-closed Subsets
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In  the following problems were listed as open: Problem 14 . 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  used Boolean rings and M. Katˇetov  used topological methods. In , a different method was used to answer the question for Hausdorff-closed spaces and in , the other two questions were answered in
the affirmative. In this paper, different proofs from those in  and  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  and Vermeer  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.
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