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 Hru\v{s}\'{a}k.