Irresolvable and submaximal spaces: Homogeneity versus σ-discreteness and new ZFC examples

An example of an irresolvable dense subspace of {0,1}c is constructed in ZFC. We prove that there can be no dense maximal subspace in a product of first countable spaces, while under Boothʹs Lemma there exists a dense submaximal subspace in [0,1]c. It is established that under the axiom of constructibility any submaximal Hausdorff space is σ-discrete. Hence it is consistent that there are no submaximal normal connected spaces. If there exists a measurable cardinal, then there are models of ZFC with non-σ-discrete maximal spaces. We prove that any homogeneous irresolvable space of non-measurable cardinality is of first category. In particular, any homogeneous submaximal space is strongly σ-discrete if there are no measurable cardinals.