Generalized submaximal spaces
type: Item
instance of: scholarly article
title: Generalized submaximal spaces
zbMATH Open document ID: 1265.54007
published in: Acta Mathematica Hungarica
publication date: 2012-02-13T00:00:00Z
review text: According to \textit{Á. Császár} [Acta Math. Hung. 96, No. 4, 351--357 (2002; Zbl 1006.54003)], a generalized topology on a set \(X\) is a family \(\mu\) of subsets of \(X\) such that \(\emptyset \in \mu\) and the arbitrary union of members of \(\mu\) belongs to \(\mu\). The elements of \(\mu\) are called \(\mu\)-open sets. In a straightforward manner, \(\mu\)-closed sets can be defined as well as the \(\mu\)-interior and the \(\mu\)-closure of an arbitrary subset \(S\subseteq X\). As a consequence, the definitions of semi-open sets and preopen sets can be extended to the context of \(\mu\)-open sets. The present paper investigates the situation where a generalized space \((X,\mu )\) is submaximal, i.e. every \(\mu\)-dense set is \(\mu\)-open. It turns out that several results from the topological setting are also valid for generalized spaces. In particular, a generalized space \((X,\mu )\) is submaximal if and only if every subset of \(X\) is \(\mu\)-locally closed.
Mathematics Subject Classification ID: 54A05
Mathematics Subject Classification ID: 54C05
zbMATH DE Number: 6006172
zbMATH Keywords: generalized submaximal space
zbMATH Keywords: generalized topology
zbMATH Keywords: generalized space
zbMATH Keywords: generalized extremally disconnected space
author: Erdal Ekici
MaRDI profile type: Publication
full work available at URL: s10474-011-0109-z
OpenAlex ID: W2085629568
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DOI: 10.1007/S10474-011-0109-Z
description: scientific article; zbMATH DE number 6006172