Almost all submaximal groups are paracompact and σ-discrete

We prove that any topological group of a non-measurable cardinality is hereditarily paracompact and strongly σ-discrete as soon as it is submaximal. Consequently, such a group is zero-dimensional. Examples of uncountable maximal separable spaces are constructed in ZFC. 0. Introduction. This paper has been motivated by several unsolved problems in general topology and topological algebra. The concepts we are mainly going to deal with are maximality and submaximality of general topological spaces introduced in [He] more than fifty years ago. Recall that a topological space X is called submaximal if every dense subset of X is open. In this paper we consider only submaximal spaces without isolated points, so “submaximal” is to be read “submaximal dense in itself ”. A dense-in-itself space X is called maximal if any strictly stronger topology on X has isolated points. Although it is not evident at first glance, every maximal space is submaximal. Let us mention that all definitions and formulations related to the topic will be given in Section 1 (Notation and terminology) or in the main text. A reader who has not got the hang of the subject can be referred to an excellent paper of van Douwen [vD] which combines detailed and transparent proofs with covering practically everything important in the theory of maximal and submaximal spaces up to the year 1990. 1991 Mathematics Subject Classification: Primary 54H11, 22A05, 54G05.