Normality in terms of distances and contractions

TL;DR

This paper investigates the concept of normality through distances and contractions, establishing its equivalence to key separation and extension properties in approach spaces, with implications for metric and topological spaces.

Contribution

It introduces a distance-based notion of normality and proves its equivalence to separation by Urysohn maps, interpolation, and extension theorems, unifying these concepts in approach spaces.

Findings

Normality is equivalent to separation by Urysohn contractive maps.

Normality corresponds to Katětov-Tong's interpolation for bounded functions.

Normality aligns with Tietze's extension theorem in classical topology.

Abstract

The main purpose of this paper is to explore normality in terms of distances between points and sets. We prove some important consequences on realvalued contractions, i.e. functions not enlarging the distance, showing that as in the classical context of closures and continuous maps, normality in terms of distances based on an appropriate numerical notion of $\gamma$-separation of sets, has far reaching consequences on real valued contractive maps, where the real line is endowed with the Euclidean metric. We show that normality is equivalent to (1) separation of $\gamma$-separated sets by some Urysohn contractive map, (2) to Kat\v{e}tov-Tong's interpolation, stating that for bounded positive realvalued functions, between an upper and a larger lower regular function, there exists a contractive interpolating map and (3) to Tietze's extension theorem stating that certain contractions defined on a subspace can be contractively extended to the whole space. The appropriate setting for these investigations is the category of approach spaces, but the results have (quasi)-metric counterparts in terms of non-expansive maps. Moreover when restricted to topological spaces, classical normality and its equivalence to separation by a Urysohn continuous map, to Kat\v{e}tov-Tong's interpolation for semicontinuous maps and to Tietze's extension theorem for continuous maps are recovered.