Almost uniform convergence
type: Item
instance of: scholarly article
title: Almost uniform convergence
zbMATH Open document ID: 0787.54019
DOI: 10.1007/BF01875883
author: V. Pereyra
published in: Periodica Mathematica Hungarica
publication date: 1994-05-19T00:00:00Z
review text: A net \((f_ n)\) of functions on a topological space \(X\) to a uniform space \((Y,{\mathcal U})\) converges almost uniformly to a function \(f\) at \(x_ 0\in X\) if for each \(U\in{\mathcal U}\) there exists a neighborhood \(W\) of \(x_ 0\) such that eventually \((f_ n(x),f(x))\in U\) for each \(x\in W\). Clearly almost uniform convergence lies between uniform convergence and quasi-uniform convergence. The author shows that if \(X\) is Hausdorff, then almost uniform convergence coincides with quasi-uniform convergence if and only if \(X\) is discrete. A paracompact Tykhonov space \(X\) is compact if and only if uniform convergence and almost convergence coincide. A Tykhonov space \(X\) is locally compact if and only if uniform convergence on compacta equals almost uniform convergence. The author also provides suitable examples. (Reviewer's remarks: In Corollary 2-3 read paracompact for compact. For related work see [\textit{A. Di Concilio} and the reviewer, Monatsh. Math. 103, 93-102 (1987; Zbl 0607.54013)] and [the reviewer, Czech. Math. J. 37(112), 608-612 (1987; Zbl 0647.54003)].
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 54A20
Mathematics Subject Classification ID: 54E15
Mathematics Subject Classification ID: 54D30
zbMATH DE Number: 508827
zbMATH Keywords: almost uniform convergence
zbMATH Keywords: quasi-uniform convergence
reviewed by: S. Naimpally
MaRDI profile type: Publication
cites work: A topology for spaces of transformations
full work available at URL: https://doi.org/10.1007/bf01875883
OpenAlex ID: W2093068058
description: scientific article; zbMATH DE number 508827