  As indicated in the title, this paper was inspired by Smithson's work on
Arzel\`a-Ascoli theorem. Up to this point, the author's interest in the topic
has led him to generalize Arzel\`a-Ascoli theorem in the context of uniform
spaces. This approach was primarily influenced by J.L. Kelley's \textit{General
topology}. Soon after, the author learnt that in 1971, R.E. Smithson carried
over Kelley's approach to the case of point-compact multifunctions (rather than
single-valued functions). Similarly, in this paper the author would like to
carry over the result, presented in his previous paper \textit{Arzel\`a-Ascoli
theorem in uniform spaces}, to the case of point-compact multifunctions.
