SUBMAXIMALITY, EXTREMAL DISCONNECTEDNESS AND GENERALIZED CLOSED SETS

ABSTRACT. In this paper, we continue the study of generalized closed sets in a topological space. In particular, we study the question when some classes of generalized closed sets coincide. A new class of spaces, the class of sg-submaximal spaces, is also introduced. Characterizations of extremally disconnected spaces and sg-submaximal spaces are established via various kinds of generalized closed sets. 1. I~vTR~DUCTI~N During the last few years the study of generalized closed sets has found consid-erable interest among general topologists. One reason is these objects are natural generalizations of closed sets. More importantly, generalized closed sets suggest some new separation axioms which have been found to be very useful in the study of certain objects of digital topology, for example, the digital line (see e.g. [S]). In [5], Dontchev summarized in a diagram the fundamental relationships between the various types of generalized closed sets. Concerning this diagram, he also posed two questions which have been answered by Cao, Ganster and Reilly in [4]. In the present paper, we take another look at Dontchev’s diagram. In doing so, we are able to characterize extremally disconnected spaces via generalized closed sets. Moreover, we introduce the notion of sg-submaximal spaces and investigate this class of spaces in terms of the Hewitt decomposition of a topological space. 1991 Mathematics Subject Classi&ation. 54A05, 54F65, 54GO5. Key words and phrases. Preclosed, sg-closed, gcr-closed, sg-submaximal, extremally