A density counterpart of the Scheepers covering property

TL;DR

The paper develops a density analogue of the Scheepers covering property by introducing $S$-separability and its weakened variants, and studies their relationship to $M$-separability. It shows that under the Near Coherence of Filters (NCF) principle, $S$-separability, $mS$-separability, and $M$-separability coincide for countable spaces, while under CH there exist countable spaces that are $mS$-separable but not $S$-separable. It connects these notions to spaces of functions via $C_p(T)$, where the three separability notions become equivalent, and analyzes FU spaces to relate countable FU spaces to $S$-separability. The results highlight the set-theoretic sensitivity of density analogues of Scheepers-type properties and illuminate their interactions with function spaces and finite-power behavior.

Abstract

We introduce a density counterpart of the Scheepers covering property $\bigcup_{\mathrm{fin}}(\mathcal O,Ω)$ and study its relations to known combinatorial density property. In particular, we show that it is equivalent to the $M$-separability under the Near Coherence of Filters principle of Blass and Weiss.

Theorems & Definitions (19)

• Theorem 1.1
• Theorem 1.3
• Theorem 1.6
• Theorem 1.8
• Definition 2.1
• Lemma 2.2
• proof
• Proposition 2.3
• proof
• Theorem 3.1