On submaximal spaces
type: Item
instance of: scholarly article
title: On submaximal spaces
zbMATH Open document ID: 0826.54002
DOI: 10.1016/0166-8641(94)00093-I
published in: Topology and its Applications
publication date: 1995-12-05T00:00:00Z
review text: ``Locally closed sets, sets open in their closure, play an enigmatic role in topology. Their appearance is often surprising and a general theory is elusive: one only needs to note that locally compact subsets of Hausdorff spaces and connected subsets of the real line have the property.'' With this introduction, the authors begin a detailed and extremely well written investigation of this property. Included is a lengthy list of references which, in combination with this article, will get a novice in the study of this property off to a great start. Some of the results are: (1) when scattered, such submaximal spaces are nodec (in the sense of van Douwen) and conversely; (2) Every countably compact Hausdorff nodec space is the free topological sum of finitely many Alexandroff compactifications of discrete spaces; (3) Pseudocompact Tychonoff submaximal spaces are scattered; (4) Pseudo-Lindelöf submaximal spaces are Lindelöf. The last section is the study of submaximal topological groups where it is shown that a pseudo-compact submaximal topological group is finite and a totally bounded submaximal topological group is countable. The authors admit that their study does not reveal all as unanswered questions start appearing early in section 1 with the first being ``Is every submaximal space \(\sigma\)-discrete?'' Another is ``Is there an infinite regular connected submaximal space?'' The reviewer wishes that every paper was written as well as this one. The writing makes it both easy and enjoyable to read. I hope that any of you read and enjoy as I did.
Mathematics Subject Classification ID: 54A10
Mathematics Subject Classification ID: 54D30
zbMATH DE Number: 792536
zbMATH Keywords: countably compact Hausdorff nodec space
zbMATH Keywords: pseudo-compact submaximal topological group
zbMATH Keywords: totally bounded submaximal topological group
author: A. V. Arhangel'skii
author: Pieter Collins
reviewed by: Douglas E. Cameron
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 792536