A measure of non-compactness on $T_{3\frac{1}{2}}$ spaces based on Arzelà-Ascoli type theorem

Abstract

The purpose of this paper is to generalize the measure of non-compactness for the space of continuous functions over the $T_{3\frac{1}{2}}$ space. Motivated by the generalized Arzelà-Ascoli theorem for Tichonoff space $T$ via Wallman compactifiaction $\mathrm{Wall}(T)$, we constuct a measure of non-compactness for the space $C^b(T)$. We also study some of the properties of this object and give another version of Darbo-type theorem, suitable for this particular case.