Bulletin of the Korean Mathematical Society (대한수학회보)
- Volume 20 Issue 2
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- Pages.95-97
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- 1983
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- 1015-8634(pISSN)
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- 2234-3016(eISSN)
A NOTE ON S-CLOSED SPACES
- Woo, Moo-Ha (Korea University) ;
- Kwon, Taikyun (Korea University) ;
- Sakong, Jungsook (Korea University)
- Published : 1983.10.01
Abstract
In this paper, we show a necessary and sufficient condition for QHC spaces to be S-closed. T. Thomson introduced S-closed spaces in [2]. A topological space X is said to be S-closed if every semi-open cover of X admits a finite subfamily such that the closures of whose members cover the space, where a set A is semi-open if and only if there exists an open set U such that U.contnd.A.contnd.Cl U. A topological space X is quasi-H-closed (denote QHC) if every open cover has a finite subfamily whose closures cover the space. If a topological space X is Hausdorff and QHC, then X is H-closed. It is obvious that every S-closed space is QHC but the converse is not true [2]. In [1], Cameron proved that an extremally disconnected QHC space is S-closed. But S-closed spaces are not necessarily extremally disconnected. Therefore we want to find a necessary and sufficient condition for QHC spaces to be S-closed. A topological space X is said to be semi-locally S-closed if each point of X has a S-closed open neighborhood. Of course, a locally S-closed space is semi-locally S-closed.
Keywords