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
Section
Original Articles