Ultracomplete topological spaces
type: Item
instance of: scholarly article
title: Ultracomplete topological spaces
zbMATH Open document ID: 0997.54037
DOI: 10.1023/A:1013743709044
published in: Acta Mathematica Hungarica
publication date: 2002-10-30T00:00:00Z
review text: If \(X\) is a set and \(\Sigma\) is a set of coverings of \(X\), then we can consider \((X,\Sigma)\) to be a generalized uniform space. This point of view originates from the work of \textit{J. W. Tukey} [Convergence and uniformity in topology, Princeton (1940)] who used \((X,\Sigma)\), where \(\Sigma\) fulfills some appropriate conditions, as another approach to define uniform spaces. Is now \(\phi\) a filter on \(X\), one can define \(\phi\) to be Cauchy, iff \(\forall \alpha\in \Sigma: \phi\cap\alpha\neq\emptyset\) holds. For a Tukey-uniform space \((X,\Sigma)\), \textit{H. H. Corson} [Am. J. Math. 80, 185-190 (1958; Zbl 0080.15803)] defined a filter \(\phi\) to be weakly Cauchy, iff for all \(\alpha\in \Sigma\) there exists a refining filter \(\psi\supseteq\phi\), such that \(\psi\cap\alpha\neq\emptyset\). In 1937, in his famous paper [Ann. Math. (2) 38, 823-844 (1937; Zbl 0017.42803)], \textit{E. Čech} also defined the now called Čech-complete spaces: A Tikhonov space \(X\) is said to be Čech complete, iff it is a \(G_{\delta}\)-set in one (equivalently in all) of its Hausdorff compactifications. An internal characterization of Čech complete spaces was established independently by \textit{Z. Frolík} [Czech. Math. J. 10, 359-379 (1960; Zbl 0100.18701)] and by \textit{A. V. Arkhangel'skij} [Vestn. Mosk. Univ., Ser. I 16, 37-39 (1961; Zbl 0106.15702)]: A Tikhonov space \(X\) is Čech complete, iff there exists a countable generalized uniform structure \(\Sigma=(\alpha_n)_{n\in \mathbb N}\) of open covers of \(X\) such that every \(\Sigma\)-Cauchy filter clusters in \(X\). There are some interesting topological notions which are related to Čech completeness. For Tukey-uniform spaces, \textit{N. R. Howes} [Pac. J. Math. 38, 431-440 (1971; Zbl 0202.54002)] introduced the notion of a confinally complete uniform space and proved that a uniform space is confinally complete, iff every weak Cauchy filter has a cluster point. This notion independently was also defined by \t…
Mathematics Subject Classification ID: 54D35
Mathematics Subject Classification ID: 54D45
Mathematics Subject Classification ID: 54E50
zbMATH DE Number: 1821677
zbMATH Keywords: ultracompletness
zbMATH Keywords: Čech completness
zbMATH Keywords: local completness
author: David Buhagiar
author: Iwao Yoshioka
reviewed by: Harry Poppe
MaRDI profile type: Publication
full work available at URL: a:1013743709044
OpenAlex ID: W1950976294
description: scientific article; zbMATH DE number 1821677