A new order for ideal sequential compactness

TL;DR

This paper introduces a new order on ideals related to a variant of sequential compactness in topological spaces, compares it with existing orders, and explores its implications under the Continuum Hypothesis, addressing open questions.

Contribution

It defines the preorder _{BW} on ideals, studies its properties, and establishes conditions under which _{BW} relations can be reversed for _{BW} spaces, especially for _{F_\sigma} ideals.

Findings

Under CH, the _{BW} order can be reversed for _{F_\sigma} ideals.

_{BW} is compared with the Kattov order, revealing new relationships.

Answers to open questions about the comparison of (\u0010_{W}) spaces with sequentially compact spaces.

Abstract

Let $\mathcal{I}$ be an ideal on $\omega$ and $X$ be a topological space. A sequence $(x_n)_{n\in \omega}$ in $X$ is $\mathcal{I}$-convergent if there is $x\in X$ such that $\{n\in \omega:x_n\notin U\}\in\mathcal{I}$ for every open neighborhood $U$ of $x$. We examine the following variant of sequential compactness associated with $\I$: $X$ is $\mathrm{BW}(\mathcal{I})$ if for every sequence $(x_n)_{n\in \omega}$ in $X$ there is $A\notin\mathcal{I}$ such that $(x_n)_{n\in A}$ is $\mathcal{I}$-convergent. We introduce a new preorder on ideals, denoted $\leq_{BW}$, such that $\mathcal{I}\leq_{BW}\mathcal{J}$ implies that every $\mathrm{BW}(\mathcal{J})$ space is $\mathrm{BW}(\mathcal{I})$. Our main result states that under CH the above implication can be reversed in the case of $\mathbf{F_\sigma}$ ideals $\I$ and $\J$. We compare $\leq_{BW}$ with the Kat\v{e}tov order and study the relation $\leq_{BW}$ among some well-known ideals (e.g. the van der Waerden ideal $\mathcal{W}$ consisting of all subsets of $\omega$ that do not contain arbitrary long finite arithmetic progressions). As a consequence, we answer two open questions posed by Filip\'{o}w and Tryba in [Top. App. {\textbf{178}} (2014), 438--452] concerning comparison of $\mathrm{BW}(\mathcal{W})$ with the class of sequentially compact spaces.