Compact Spaces Via p-closed Subsets

  • Terrence A. Edwards University of the District of Columbia
  • James E. Joseph Howard University
  • Myung H. Kwack Howard University
  • Bhamini M.P. Nayar Morgan Sate University

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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Author Biographies

Terrence A. Edwards, University of the District of Columbia
Department of Mathematics
James E. Joseph, Howard University
Deaprtment of Mathematics
Myung H. Kwack, Howard University
Department of Mathematics

References

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[JN2] J. E. Joseph and B. M. P. Nayar, A Hausdorff(Urysohn)[Regular] Space in Which All of Its Closed sets are Hausdorff-closed(Urysohn-closed)[Regular-closed] Is Compact, Journal of Advanced Studies in Topology (To appear).

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Published
2014-02-07
How to Cite
Edwards, T. A., Joseph, J. E., Kwack, M. H., & Nayar, B. M. (2014). Compact Spaces Via p-closed Subsets. Journal of Advanced Studies in Topology, 5(2), 8-12. https://doi.org/10.20454/jast.2014.743
Section
Original Articles

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