Some Properties Of Proximal Homotopy Theory

TL;DR

This paper explores the integration of nearness and proximity concepts into homotopy theory, examining how proximity spaces influence homotopy equivalence and related properties.

Contribution

It introduces the application of proximity space concepts to homotopy theory, expanding the understanding of closeness in topological and metric contexts.

Findings

Proximity spaces can be used to analyze homotopy properties.

Descriptive proximity spaces extend the application of nearness in homotopy theory.

The study links nearness concepts with classical homotopy notions.

Abstract

Nearness theory comes into play in homotopy theory because the notion of closeness between points is essential in determining whether two spaces are homotopy equivalent. While nearness theory and homotopy theory have different focuses and tools, they are intimately connected through the concept of a metric space and the notion of proximity between points, which plays a central role in both areas of mathematics. This manuscript investigates some concepts of homotopy theory in proximity spaces. Moreover, these concepts are taken into account in descriptive proximity spaces.