# 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.

### 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

[Bo] N. Bourbaki, Espaces Minimax et Espaces Compl$\grave e$tement s\'epar\'es, C. R. Acad. Sci. Paris 212 (1941), 215-218.

[BPS] M. P. Berri, J. R. Porter and R. M. Stephenson, Jr, A Survey of Minimal Topological Spaces, General Topology and Its Relations To Modern Analysis and Algebra,Proc. Conf. Kanpur (1968) (Academia Prague (1971), 93-114.

[BS] M. P. Berri and R. H. Sorgenfrey, Minimal Regular Spaces, Proc. Amer. Math Soc. 14 (1963), 454-458.

[EJK] M. S. Espelie, J. E. Joseph and M. H. Kwack, Application of the $u$-closure Operator, Proc. Amer. Math. Soc. 83 (1981), 167-174.

[G] M. Girou, Properties of Locally $H$-Closed Spaces, Proc. Amer. Math. Soc. Vol. 113 N0. 1 (1991),287 - 295.

[H1] L. L. Herrington, Characterizations of Urysohn-closed Spaces, Proc. Amer. Math. Soc. 53 (1976), 435-439.

[H2] L. L. Herrington, Characterizations of Regular-closed Spaces, Math. Chronicle 5 (1977), 168 -178.

[He1] H. Herrlich, $T_v$-Abgeschlossenheit und $T_v$-Minimalit\"at, Mah. Z. 88 (1965), 285-284.

[He2] H. Herrlich, Regular-closed, Urysohn-closed and Completely Hausdorff-closed Spaces, Proc. Amer. Math. Soc. 26 (1970), 695-698.

[J1] J. E. Joseph, Urysohn-closed and Minimal Urysohn Spaces, Proc. Amer. Math. Soc. 68 (2) (1978), 235
-242.

[J2] J. E. Joseph, Regular-closed and Minimal Regular Spaces, Canad. Math. Bull. Vol. 22 (4) (1979), 491 -497.

[JKN] J E. Joseph, M. H. Kwack, B. M. P. Nayar, Weak continuity Forms, Graph conditions, and Applications, Scientiae Mathematicae {\bf 2} (1999), 65-88.

[JN1] J. E. Joseph and B. M. P. Nayar, A New Proof for a Theorem of Stone and Kat\v etov, International Journal of Pure and Applied Mathematics, Vol. 89 No.2 (2013) 287-288.

[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).

[K] M. Kat\v etov, Uber H-bgeschlossene und Bikompakte Raume, Casopis Pest. Mat. 69 (1940), 36-49 (German).

[S] M. H. Stone, {\it Applications of the Theory of Boolean Rings to General Tology}, Trans. Amer. Math. Soc. 41 No. 3 (1937), 375-481.

[Sc] C. T. Scarborough, Minimal Urysohn Spaces, Pacific J. Math. 27 (1968), 611-617.

[V] N. V. Veli\u cko, $H$-Closed Topological Spaces, Mat. Sb. Vol. 70 (112) (1966), 98 - 112: Amer. Math. Soc. Transl. Vol. (2) 78 (1968), 103-118.

[Ve] J. Vermeer, Closed Subspaces of H-closed Spaces, Pacific J. Math. 118 N0. 1 (1985), 229-247.
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
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Section
Original Articles