PROCEEDINGS of the AMERICAN MATHEMATICAL SOCIETY Volume 74, Number 2, May 1979 A NOTE ON S-CLOSED SPACES ASHA MATHUR1 Abstract. A topological space X is said to be S-closed if and only if for every semi-open cover of X there exists a finite subfamily such that the union of their closures cover X. For a Hausdorff space, the concept of S-closed is shown to be equivalent to the concept of extremally disconnect- ed and nearly compact. Further it has been shown that EDH-closed spaces are precisely S-closed Hausdorff spaces. T. Thompson introduced and studied S-closed spaces in [6]. A space X is said to be S-closed if every semi-open cover of X admits a finite subfamily, the closures of whose members cover the space, where a set S is semi-open if and only if there exists an open set V such that V c A c V. Since every regularly closed set is semi-open and the closure of semi-open sets is regularly closed the following result is immediate. Theorem 1. The following are equivalent for a topological space X: (a) X is S-closed. (b) Every regularly closed cover has a finite subcover. (c) Every family of regularly open sets having the finite intersection property has nonempty intersection. Since every S-closed Hausdorff space is extremally disconnected [6, Theorem 7] and in extremally disconnected spaces regularly closed sets coincide with clopen sets, the following corollary follows from the above theorem. Corollary 2. The following are equivalent for a Hausdorff space X: (a) X is S-closed. (b) Every regularly closed cover has a finite disjoint clopen refinement. The main result obtained by T. Thompson is that for compact Hausdorff or compact regular spaces, the concept of S-closed is equivalent to the concept of extremally disconnected. Since in a regular space every open cover can be refined by a regularly closed cover it is immediate from Theorem 1(b) Received by the editors December 15, 1977 and, in revised form, August 19, 1978. AMS (MOS) subject classifications (1970). Primary 54DXX, 54D20, 54D30;Secondary 54G05. Key words and phrases. S-closed, extremally disconnected, nearly compact, EDH-closed. 'The author gratefully acknowledges the financial support given by the Association of Commonwealth Universities by awarding the Commonwealth Scholarship to the author at the University of St. Andrews. Also, she would like to express her gratitude to Dr. Roy Dyckhoff of the University of St. Andrews for the valuable discussions she had with him during the preparation of this note. © 1979 American Mathematical Society 0002-9939/79/0000-0229/S01.75 350 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use