Very I-favorable spaces

TL;DR

This paper characterizes very I-favorable spaces as almost limit spaces of specific inverse systems and links Tychonoff very I-favorable spaces with d-openly generated spaces.

Contribution

It provides a new characterization of very I-favorable spaces using inverse systems and establishes their relation to d-openly generated spaces.

Findings

Very I-favorable spaces are characterized as almost limit spaces of $\sigma$-complete inverse systems.

Tychonoff very I-favorable spaces with respect to co-zero sets are exactly the d-openly generated spaces.

The paper bridges the concepts of inverse systems, d-open maps, and I-favorability in topological spaces.

Abstract

We prove that a Hausdorff space $X$ is very $\mathrm I$-favorable if and only if $X$ is the almost limit space of a $\sigma$-complete inverse system consisting of (not necessarily Hausdorff) second countable spaces and surjective d-open bonding maps. It is also shown that the class of Tychonoff very $\mathrm I$-favorable spaces with respect to the co-zero sets coincides with the d-openly generated spaces.