On \(\theta\)-connectedness and \(\theta\)-closure spaces
type: Item
instance of: scholarly article
title: On \(\theta\)-connectedness and \(\theta\)-closure spaces
zbMATH Open document ID: 1009.54022
DOI: 10.1016/S0166-8641(01)00179-1
author: Mila Mršević
author: Dimitrije Andrijević
published in: Topology and its Applications
publication date: 2002-09-17T00:00:00Z
review text: If \((X,{\mathcal T})\) is a topological space and \(A\subseteq X\), then the \(\theta\)-closure of \(A\), denoted by cl\(_\theta(A)\) is the set \(\{x:\) each closed neighbourhood of \(x\) meets \(A\}\). A set \(C\) is \(\theta\)-closed if \(C=\text{ cl}_\theta(C)\); the \(\theta\)-closed sets form the closed sets of a topology called the \(\theta\)-topology, denoted here by \({\mathcal T}_\theta\). The \(\theta\)-closure operator is not idempotent but is a Čech closure operator. This paper studies properties of the \(\theta\)-topology and the semiregularization topology \({\mathcal T}_s\) (the topology generated by the regular open sets). Among the results it is shown that local connectedness (respectively, compactness, normality) of \((X,{\mathcal T})\) implies the corresponding condition for \((X,{\mathcal T}_s)\) which in turn implies that \((X,{\mathcal T}_\theta)\) is locally connected (respectively, compact, normal).
Mathematics Subject Classification ID: 54D05
Mathematics Subject Classification ID: 54A05
Mathematics Subject Classification ID: 54A10
Mathematics Subject Classification ID: 54D10
Mathematics Subject Classification ID: 54D15
zbMATH DE Number: 1801760
zbMATH Keywords: \(\theta\)-closure
zbMATH Keywords: \(\delta\)-closure
zbMATH Keywords: neighbourhood space
zbMATH Keywords: semi-regularization topology
reviewed by: Richard G. Wilson
MaRDI profile type: Publication
cites work: Q5522002
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cites work: Some Properties of θ-Closure
cites work: Q5837221
cites work: Q3946771
cites work: Q3844854
cites work: Q5636020
cites work: Q3943648
cites work: SOME PROPERTIES OF THE SPACE 2XFOR A TOPOLOGICALR0-SPACE
cites work: Q5186346
cites work: Q4186971
cites work: Q5613067
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description: scientific article; zbMATH DE number 1801760