Submaximal spaces and cardinal invariants

Abstract

We give a combinatorial characterization of countable submaximal subspaces of $2^{\kappa }$. Using a parametrized version of Mathias forcing, we prove that there exists a countable submaximal subspace of $2^{{\omega }_1}$ whilst $\mathfrak{c}={\omega }_2$. Combining this with previous results, we construct a disjointly tight countable irresolvable space of weight $<\mathfrak{c}$, answering a question of Bella and Hrušák.

arXiv comment11 pages