Lightly compact spaces and infra $H$-closed spaces

Abstract

We show that a topological space is lightly compact if and only if each countable open filter base has a cluster point. This gives a direct connection with H-closed spaces, and suggests the definition of infra H-closed spaces as those topological spaces whose continuous images in first countable Hausdorff spaces are closed. These properties are distinct and provide another characterization of pseudocompact T3 spaces. In [1] light compactness was used to characterize pseudocompact completely regular spaces, and in [7] it was used in obtaining several filter characterizations of quasi-regular Baire spaces. In this paper we obtain a filter characterization of light compactness and several consequences including a closedness property related to H-closure. A space X is called H-closed if for any Hausdorff space Y and any continuous /: X Y, / [X] is closed in Y. A space X is called lightly compact if every locally finite collection of open subsets of X is finite. It is well known (see [6]) that H-closed spaces can be characterized as those for which every open cover contains a finite subcollection whose union is dense. There is a striking similarity between this and the following characterization of light compactness which appears in [3]. A space is lightly compact if and only if every countable open cover contains a finite subcollection whose union is dense. We now obtain another characterization of light compactness. By an open filter base in a space we will mean a nonempty collection of nonempty open sets such that the intersection of any two elements of the collection contains an element of the collection. If Jf is an open filter base in a space X, we will say that a point x 6 X is a cluster point of ff if x f nlclx(U): U ET. Received by the editors March 10, 1973 and, in revised form, February 10, 1974. AMS (MOS) subject classifications (1970). Primary 54D30; Secondary 54D20, 54D25.