Arzelà-Ascoli theorem via Wallman 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-Čech compactification. The main result of the paper is a new form of Arzelà-Ascoli theorem, which introduces the concept of equicontinuity along $ω$-ultrafilters.