Abstract
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 Booth's 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.
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