Infinite dimensional sequential compactness: Sequential compactness based on barriers

TL;DR

This paper generalizes the concept of sequential compactness using barriers in infinite-dimensional spaces, constructing new examples and analyzing properties related to barriers and cardinal invariants.

Contribution

It introduces a generalized notion of sequential compactness via barriers, constructs specific spaces with distinct barrier-based compactness properties, and explores associated cardinal invariants.

Findings

Constructed spaces that are -sequentially compact but not -sequentially compact under certain conditions.

Identified classes of spaces that are -sequentially compact for all barriers .

Derived results on cardinal invariants related to barriers.

Abstract

We introduce a generalization of sequential compactness using barriers on $\omega$ extending naturally the notion introduced in [W. Kubi\'{s} and P. Szeptycki, On a topological Ramsey theorem, \emph{Canad. Math. Bull.}, 66 (2023), {156}--{165}]. We improve results from [C. Corral and O. Guzm{\'a}n and C. L{\'o}pez-Callejas, High dimensional sequential compactness, \emph{Fund. Math.}] by building spaces that are $\mathcal{B}$-sequentially compact but no $\mathcal{C}$-sequentially compact when the barriers $\mathcal{B}$ and $\mathcal{C}$ satisfy certain rank assumption which turns out to be equivalent to a Kat\v{e}tov-order assumption. Such examples are constructed under the assumption $\mathfrak{b} =\mathfrak{c}$. We also exhibit some classes of spaces that are $\mathcal{B}$-sequentially compact for every barrier $\mathcal{B}$, including some classical classes of compact spaces from functional analysis, and as a byproduct we obtain some results on angelic spaces. Finally we introduce and compute some cardinal invariants naturally associated to barriers.