Abstract
In [BPS] the following problems were listed as open: Problem 14.[B] 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 Urysohn-closed compact? To answer the question for Hausdorff-closed spaces in the affirmative, M. H. Stone [S] used Boolean rings and M. Kat\v etov [K] used topological methods. In [JN1], a different method was used to answer the question for Hausdorff-closed spaces and in [JN2], the other two questions were answered in the affirmative. In this paper, different proofs from those in [JN1] and [JN2] are given answering all of these questions. An affirmative answer is also given to a question posed by Girou [G] and Vermeer [Ve] 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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