Short note on Smithson's paper concerning Arzel\`a-Ascoli theorem for multifunctions

TL;DR

This paper extends the Arzelà-Ascoli theorem to point-compact multifunctions within uniform spaces, building on Smithson's and Kelley's foundational work, and aims to generalize previous single-valued function results.

Contribution

It generalizes the Arzelà-Ascoli theorem for multifunctions in uniform spaces, expanding upon prior single-valued function theorems and Smithson's earlier work.

Findings

Extension of Arzelà-Ascoli theorem to multifunctions

Application of uniform space framework

Connection to Smithson's and Kelley's approaches

Abstract

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.