New examples in the study of selectively separable spaces

TL;DR

The paper addresses the problem of distinguishing wH-separable, mH-separable, and H-separable properties in countable spaces. It introduces a novel recursive construction on the rationals by building a chain of clopen bases $\tau_\alpha$ and employing $I(q,f)$-type sets together with elementary submodels of $H(\mathfrak c^+)$ to control convergence and density across the hierarchy. The main results include a ZFC example of a countable regular 0-dimensional space that is wH-separable but not H-separable, along with Fréchet-Urysohn instances under weaker hypotheses and prescribed values of cardinal invariants such as $\mathfrak b$ and $\mathfrak p$. Additionally, a separate construction yields a countable space that is mH-separable but not H-separable, clarifying relationships among these selective separability notions in countable topological spaces.

Abstract

The property of being selectively separable is well-studied and generalizations such as H-separable and wH-separable have also generated much interest. Bardyla, Maesano, and Zdomskyy proved from Martin's Axiom that there are countable regular wH-separable spaces that are not H-separable. We prove there is a ZFC example. Their example was also Fréchet-Urysohn, and we produce two additional examples from weaker assumptions.

Theorems & Definitions (33)

• Theorem 1
• Theorem 2: $\mathfrak b = \mathfrak c$
• Theorem 3: $\mathfrak p = \mathfrak b$
• Corollary 4: $\mathfrak c \leq \omega_2$
• Definition 5
• Definition 6
• Proposition 7
• Proposition 8
• proof
• Proposition 9