Selections, games and metrisability of manifolds

TL;DR

This paper explores how certain selection principles relate to the metrisability and separability of manifolds, establishing equivalences that connect topological properties with these principles.

Contribution

It identifies specific selection principles that are equivalent to metrisability and separability in manifolds, clarifying their topological significance.

Findings

$ extsf{S}_{fin}( ext{K}, ext{O})$, $ extsf{S}_{fin}( ext{Ω}, ext{Ω})$, and $ extsf{S}_{fin}( ext{Λ}, ext{Λ})$ characterize metrisability.

$ extsf{S}_1( ext{D}, ext{D})$ characterizes separability.

Selection principles can serve as topological characterizations of manifold properties.

Abstract

In this note we relate some selection principles to metrisability and separability of a manifold. In particular we show that $\sf{S}_{fin}(\mathcal K,\mathcal O)$, $\sf{S}_{fin}(\Omega,\Omega)$ and $\sf{S}_{fin}(\Lambda,\Lambda)$ are each equivalent to metrisability for a manifold, while $\sf{S}_1(\mathcal D,\mathcal D)$ is equivalent to separability for a manifold.