# Compactness via $$\theta$$-closed and $$\theta$$-rigid 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 [8], Girou proved that a Hausdorff-closed space is Urysohn if and only if each Hausdorff-closed subset is $$\theta$$-closed. In this paper the following equivalences are proved for minimal Hausdorff spaces $$X$$:

(1) $$X$$ is compact,
(2) Each Hausdorff-closed subset of $$X$$ is $$\theta$$-closed,
(3) Each Hausdorff-closed subset of $$X$$ is $$\theta$$-rigid,
(4) Each minimal Hausdorff subset of $$X$$ is $$\theta$$-closed,
(5) Each minimal Hausdorff subset of $$X$$ is $$\theta$$-rigid,
(6) Each regular-closed subset of $X$ is Hausdorff-closed.

In addition, the followimg equivalences are proved for Hausdorff spaces $$X$$:

(1) $$X$$ is compact,
(2) $$X$$ is Hausdorff-closed (Urysohn-closed) [regular-closed] and each closed subset is $$\theta$$-closed ($$u$$-closed) [$$s$$-closed],
(3) $$X$$ is Hausdorff-closed (Urysohn-closed) [regular-closed] and each closed subset is $$\theta$$-rigid ($$u$$-rigid) [$$s$$-rigid],
(4)  $$X$$ is Urysohn-closed  (regular-closed) and each closed subset is $$\theta$$-rigid,
(5)  $$X$$ is regular-closed and each closed subset is $$u$$-rigid.

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Published
2014-07-23
How to Cite
Edwards, T. A., Joseph, J. E., Kwack, M. H., & Nayar, B. M. (2014). Compactness via $$\theta$$-closed and $$\theta$$-rigid subsets. Journal of Advanced Studies in Topology, 5(3), 28-34. https://doi.org/10.20454/jast.2014.827
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Original Articles