Singular precalibers for topological products

TL;DR

This paper proves that for singular cardinals with uncountable cofinality, the property of having a certain precaliber is preserved under taking powers and certain products of topological spaces.

Contribution

It establishes that singular precalibers are maintained in topological powers and specific products, extending understanding of precaliber behavior in topology.

Findings

Powers of spaces with precaliber κ also have precaliber κ.

Certain products of spaces with precaliber κ retain precaliber κ.

Results apply to singular cardinals with uncountable cofinality.

Abstract

In this paper we prove that if $\kappa$ is a singular cardinal with uncountable cofinality, then every power of a given topological space with precaliber $\kappa$ has precaliber $\kappa$ as well. Furthermore, if $\{X_\alpha : \alpha<\lambda\}$ is a family of topological spaces with precaliber $\kappa$ and $\lambda<cf(\kappa)$, then $\kappa$ is a precaliber for the topological product $\prod\{X_\alpha : \alpha<\lambda\}$.