Arzelà-Ascoli theorem via the Wallman compactification.

In the paper, we recall the Wallman compactification of a Tychonoff space <italic>T</italic> (denoted by Wall(<italic>T</italic>)) and the contribution made by Gillman and Jerison. Motivated by the Gelfand-Naimark theorem, we investigate the homeomorphism between <italic>Cb</italic>(<italic>T</italic>), the space of continuous and bounded functions on <italic>T</italic>, and <italic>C</italic>(Wall(<italic>T</italic>)), the space of continuous functions on the Wallman compactification of <italic>T</italic>. Along the way, we attempt to justify the advantages of the Wallman compactification over other manifestations of the Stone-Čech compactification. The main result of the paper is a new form of the Arzelà-Ascoli theorem, which introduces the concept of equicontinuity along <italic>ω</italic>-ultrafilters. [ABSTRACT FROM AUTHOR]