Quotient compact images of metric spaces, and related matters
type: Item
instance of: scholarly article
title: Quotient compact images of metric spaces, and related matters
zbMATH Open document ID: 0994.54015
DOI: 10.1016/S0166-8641(01)00145-6
author: Y. Ikeda
author: Chuan Liu
author: Y. Tanaka
published in: Topology and its Applications
publication date: 2002-08-22T00:00:00Z
review text: Some characterizations for the quotient compact images of metric spaces are obtained by means of weak bases. In this paper, the following notion of \(\sigma\)-strong networks as a generalization of development in developable spaces is given: Let \(\{{\mathcal C}_n:n \in\mathbb{N}\}\) be a sequence of covers of a space \(X\) such that \({\mathcal C}_{n+1}\) refines \({\mathcal C}_n\) for each \(n\in\mathbb{N}\), then \(\bigcup\{{\mathcal C}_n:n \in\mathbb{N}\}\) is called a \(\sigma\)-strong network for \(X\) if \(\{st(x,{\mathcal C}_n):n \in\mathbb{N}\}\) is a local network at \(x\) in \(X\). New characterizations for quotient compact images of metric spaces by means of \(\sigma\)-strong networks are obtained. The main theorem is that the following are equivalent for a Hausdorff space \(X\): (1) \(X\) is a sequence-covering, quotient compact-image of a metric space; (2) \(X\) has a point-regular weak base; (3) \(X\) is a sequential space with a \(\sigma\)-point-finite strong cs-network. Finally, the authors pose an interesting question: For a sequential space \(X\) with a point-regular \(\text{cs}^*\)-network, characterize \(X\) by means of a nice image of a metric space.
Mathematics Subject Classification ID: 54C10
Mathematics Subject Classification ID: 54E40
Mathematics Subject Classification ID: 54D55
zbMATH DE Number: 1787562
zbMATH Keywords: quotient map
zbMATH Keywords: compact map
zbMATH Keywords: sequence-covering map
zbMATH Keywords: \(\sigma\)-strong network
zbMATH Keywords: weak base
zbMATH Keywords: sequential space
reviewed by: Shou Lin
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1787562