Publication
Title
Normality in terms of distances and contractions
Author
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 Katetov-Tong's insertion, 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 Katetov-Tong's insertion for semicontinuous maps and to Tietze's extension theorem for continuous maps are recovered. (C) 2018 Elsevier Inc. All rights reserved.
Language
English
Source (journal)
Journal of mathematical analysis and applications. - New York, N.Y.
Publication
New York, N.Y. : 2018
ISSN
0022-247X
Volume/pages
461 :1 (2018) , p. 74-96
ISI
000426141100007
Full text (Publisher's DOI)
Full text (open access)
Full text (publisher's version - intranet only)
UAntwerpen
Faculty/Department
Research group
Publication type
Subject
Affiliation
Publications with a UAntwerp address
External links
Web of Science
Record
Identification
Creation 29.03.2018
Last edited 29.11.2021
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