Submaximal spaces and cardinal invariants

TL;DR

This paper characterizes countable submaximal subspaces of $2^\kappa$, introduces a forcing method to construct such spaces under certain set-theoretic assumptions, and answers a question about irresolvable spaces.

Contribution

It provides a combinatorial characterization of countable submaximal subspaces and constructs new examples of such spaces using forcing, advancing understanding of their properties.

Findings

Existence of a countable submaximal subspace of $2^{\omega_1}$ with $\mathfrak{c}=\omega_2$

Construction of a disjointly tight countable irresolvable space of weight less than $\mathfrak{c}$

Answering a question of Bella and Hrušák about irresolvable spaces.

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.