  In the paper, we generalize the Arzel\`a-Ascoli theorem in the setting of
uniform spaces. At first, we recall well-known facts and theorems coming from
monographs of Kelley and Willard. The main part of the paper introduces the
notion of extension property which, similarly as equicontinuity, equates
different topologies on $C(X,Y)$. This property enables us to prove the
Arzel\`a-Ascoli theorem for uniform convergence. The paper culminates with
applications, which are motivated by Schwartz's distribution theory. Using the
Banach-Alaoglu-Bourbaki theorem, we establish relative compactness of subfamily
of $C(\mathbb{R},\mathcal{D}'(\mathbb{R}^n))$.
