Compactness of \(H(i)\), \(U(i)\), \(R(i)\) spaces via filters

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Terrence A. Edwards
James E. Joseph
Myung H. Kwack
Bhamini M.P. Nayar

Abstract

Recently, filters were applied to give affirmative answers to two long-standing questions [1]. It was proved that a topological space is compact if and only if each closed subset is Hausdorff-closed (Urysohn-closed) [regular-closed]. As an improvement, it was established that a Hausdorff-closed (Urysohn-closed) [regular-closed] space is compact if and only if each closed subset is an \(H\)-set (a \(U\)-set) [an \(R\)-set] [2] (See AMS Mathematical reviews MR 3191275, MR 3112925). Stone, in 1937 [3], using Boolean rings and transfinite induction, proved that a Hausdorff space $X$ is compact if and only if each closed subset of \(X\) is Hausdorff-closed. In 1940, Katetov [4] gave a topological proof. Topological methods are used in this paper to generalize these results to non-Hausdorff (non-Urysohn) [non-regular] spaces. Stephenson (Scarboorough and Stone) established in [5] ([6]) that a countable minimal Urysohn (minimal regular) space is compact. It is shown here that every countable Urysohn-closed (regular-closed) space is compact. It is shown in [7] that a Hausdorff-closed (Urysohn-closed) [regular-closed] space is compact if and only if each closed subset is paracompact (metacompact). It is proved here that a Hausdorff-closed (Urysohn-closed) [regular-closed] space is compact if and only if it is Lindeloff of (countably compact ) [normal], and filters are utilized to generalize all of these results to \(H(i)\) (\(U(i)\)) [\(R(i)\)] spaces.

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How to Cite
Edwards, T. A. ., Joseph, J. E. ., Kwack, M. H. ., & Nayar, B. M. . (2022). Compactness of \(H(i)\), \(U(i)\), \(R(i)\) spaces via filters. Journal of Advanced Studies in Topology, 7(1), 37–44. Retrieved from http://m-sciences.com/index.php/jast/article/view/193
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Research Articles