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

## TL;DR

This paper introduces a new measure of non-compactness for continuous function spaces over T_{3.5} spaces, extending classical theorems using Wallman compactification to analyze compactness properties.

## Contribution

It generalizes the measure of non-compactness for C^b(T) spaces based on the generalized Arzelà-Ascoli theorem and Wallman compactification, providing new theoretical tools.

## Key findings

- Developed a new measure of non-compactness for C^b(T)
- Analyzed properties of the measure in T_{3.5} spaces
- Presented a Darbo-type theorem adapted to this context

## 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 Arzel\`a-Ascoli theorem for Tichonoff space $T$ via Wallman compactifiaction $\operatorname{Wall}(T)$, 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.