Uniform Powers of Compacta and the Proximal Game

TL;DR

This paper investigates the properties of countable uniform powers of compact spaces, proving they preserve proximality in certain cases and exploring related topological properties and open questions.

Contribution

It solves an open problem by showing countable uniform powers of compact proximal spaces are proximal, and explores related topological properties and questions.

Findings

Countable uniform power of compact proximal spaces is proximal.

Countable uniform power of a Corson compactum is collectionwise normal, countably paracompact, and Fréchet-Urysohn.

Results on first countability, realcompactness, and semi-proximal spaces.

Abstract

The countable uniform power (or uniform box product) of a uniform space $X$ is a special topology on ${}^{\omega}X$ that lies between the Tychonoff topology and the box topology. We solve an open problem posed by P. Nyikos showing that if $X$ is a compact proximal space then the countable uniform power of $X$ is also proximal (although it is not compact). By recent results of J. R. Bell and G. Gruenhage this implies that the countable uniform power of a Corson compactum is collectionwise normal, countably paracompact and Fr\'echet-Urysohn. We also give some results about first countability, realcompactness in countable uniform powers of compact spaces and explore questions by P. Nyikos about semi-proximal spaces.