  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.
