REALCOMPACTNESS IN MAXIMAL AND SUBMAXIMAL SPACES. FERNANDO HERNÁNDEZ-HERNÁNDEZ, OLEG PAVLOV,

Abstract. We study realcompactness in the classes of submaxi-mal and maximal spaces. It is shown that a normal submaximal space of cardinality less than the first measurable is realcompact. ZFC examples of submaximal not realcompact and maximal not realcompact spaces are constructed. These examples answer ques-tions posed in [1] and generalize some results from [4]. 0. Introduction The class of submaximal spaces was introduced by N. Bourbaki in Topologie Générale. We recall that a subset A of a space X is locally closed if A is open in its closure in X or, equivalently, is the intersection of an open subset and a closed subset of X. Definition 1. A space X is a submaximal space if every subset of