On M-separability of countable spaces and function spaces

TL;DR

This paper investigates M-separability and related combinatorial separability properties in countable and function spaces, revealing set-theoretic conditions affecting their closure properties and existence of maximal spaces.

Contribution

It demonstrates that under certain set-theoretic hypotheses, the class of selectively separable spaces is not closed under finite products and shows the non-existence of maximal M-separable countable spaces in specific models.

Findings

b=d hypothesis implies non-closure under finite products

No maximal M-separable countable space exists in certain models

Answers to open questions by Bella, Bonanzinga, Matveev, and Tkachuk

Abstract

We study M-separability as well as some other combinatorial versions of separability. In particular, we show that the set-theoretic hypothesis b=d implies that the class of selectively separable spaces is not closed under finite products, even for the spaces of continuous functions with the topology of pointwise convergence. We also show that there exists no maximal M-separable countable space in the model of Frankiewicz, Shelah, and Zbierski in which all closed P-subspaces of w^* admit an uncountable family of nonempty open mutually disjoint subsets. This answers several questions of Bella, Bonanzinga, Matveev, and Tkachuk.