Realcompactness in maximal and submaximal spaces

We study realcompactness in the classes of submaximal 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 questions posed in [O.T. Alas, M. Sanchis, M.G. Tkačenko, V.V. Tkachuk, R.G. Wilson, Irresolvable and submaximal spaces: homogeneity versus σ-discreteness and new ZFC examples, Topology Appl. 107 (3) (2000) 259–273] and generalize some results from [D.P. Baturov, On perfectly normal dense subspaces of products, Topology Appl. 154 (2) (2007) 374–383].