Arzel\`a-Ascoli theorem via Wallman compactification

TL;DR

This paper introduces a new form of the Arzelà-Ascoli theorem using Wallman compactification, highlighting its advantages and exploring the homeomorphism between bounded continuous functions on a space and its Wallman compactification.

Contribution

It presents a novel version of the Arzelà-Ascoli theorem based on equicontinuity along d-ultrafilters, emphasizing the benefits of Wallman compactification.

Findings

Established a homeomorphism between BC(T, dr) and BC(Wall(T), dr).

Proposed a new form of Arzele0-Ascoli theorem with d-ultrafilter equicontinuity.

Compared Wallman compactification with Stone-czech{}ech compactification.

Abstract

In the paper, we recall the Wallman compactification of a Tychonoff space $T$ (denoted by $\text{Wall}(T)$) and the contribution made by Gillman and Jerison. Motivated by the Gelfand-Naimark theorem, we investigate the homeomorphism between $BC(T,\mathbb{R})$ and $BC(\text{Wall}(T),\mathbb{R})$. Along the way, we attempt to justify the advantages of Wallman compactification over other manifestations of Stone-\v{C}ech compactification. The main result of the paper is a new form of Arzel\`a-Ascoli theorem, which introduces the concept of equicontinuity along $\omega$-ultrafilters.