Almost maximal spaces
type: Item
instance of: scholarly article
title: Almost maximal spaces
zbMATH Open document ID: 1110.22002
published in: Topology and its Applications
publication date: 2007-02-09T00:00:00Z
review text: A topological space \(X\) is called almost maximal if it is without isolated points and for every \(x \in X\) there are only finitely many ultrafilters on \(X\) converging to \(x\). We associate with every countable regular homogeneous almost maximal space \(X\) a finite semigroup \(ULT(X)\) so that if \(X\) and \(Y\) are homeomorphic, \(Ult(X)\) and \(Ult(Y)\) are isomorphic. Semigroups \(Ult(X)\) are projectives in the category \({F}\) of finite semigroups. These are bands decomposing into a certain chain of rectangular components. Under Martin's axiom, for each projective \(S\) in \({F}\), there is a countable almost maximal topological group \(G\) with \(Ult(G)\) isomorphic to \(S\). The existence of a countable almost maximal topological group cannot be established in ZFC. However, there are in ZFC countable regular homogeneous almost maximal spaces \(X\) with \(Ult(G)\) being a chain of idempotents. Finally a number of interesting open questions are given at the end of the paper.
Mathematics Subject Classification ID: 22A30
Mathematics Subject Classification ID: 54H11
Mathematics Subject Classification ID: 20M15
Mathematics Subject Classification ID: 54A05
zbMATH DE Number: 5123617
zbMATH Keywords: almost maximal space
zbMATH Keywords: topological group
zbMATH Keywords: Stone-Čech compactification
zbMATH Keywords: ultrafilter semigroup
zbMATH Keywords: projective
zbMATH Keywords: absolute coretract
author: Yevhen Zelenyuk
MaRDI profile type: Publication
full work available at URL: j.topol.2006.05.001
OpenAlex ID: W2002766455
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DOI: 10.1016/J.TOPOL.2006.05.001
description: scientific article; zbMATH DE number 5123617