Generalized open sets
type: Item
instance of: scholarly article
title: Generalized open sets
zbMATH Open document ID: 0924.54003
DOI: 10.1023/A:1006582718102
author: Ákos Császár
published in: Acta Mathematica Hungarica
publication date: 1999-10-31T00:00:00Z
review text: Various kinds of generalized open sets are studied together in this article. The author considers monotone mappings \(\gamma:\exp X\to \exp X\) (monotone in the sense that \(A\subset B\) implies \(\gamma(a)\subset \gamma(B)\)), and develops properties common to several classes of generalized open sets. A subset \(A\) of \(X\) is called \(\gamma\)-open iff \(A\subset\gamma A\), and, it is \(\gamma\)-closed iff \(X-A\) is \(\gamma\)-open. If \(X\) is a topological space and \(i\) is the interior map (i.e., \(iA=\text{int} A\)), then the \(i\)-open sets are the open sets in the given topology. Definitions of generalized open sets can then be given in terms of the appropriate \(\gamma\). The preopen sets of \textit{A. S. Mashhour, M. E. Abd El-Monsef}, and \textit{S. N. El-Deep} [Proc. Math. Phys. Soc. Egypt. 53, 47-53 (1982; Zbl 0571.54011)] become the \(\gamma\)-open sets for \(\gamma=ic\), where \(c\) is the closure map; the semi-open sets of \textit{N. Levine} [Am. Math. Mon. 70, 36-41 (1963; Zbl 0113.16304)] are the \(\gamma\)-open sets for \(\gamma=ici\). Other generalizations such as \(\alpha\)-open and \(\beta\)-open can be couched in analogous terms. Theorems involving \(\gamma\)-open sets will then apply automatically to each of these concepts; i.e., Proposition 1.1: Any union of \(\gamma\)-open sets is \(\gamma\)-open. The author continues with studies of intersections of \(\gamma\)-open sets, \(\gamma\)-open sets in subspaces, locally \(\gamma\)-closed sets, and \(\gamma\)-regular sets, with the natural definitions of these concepts.
Mathematics Subject Classification ID: 54A05
zbMATH DE Number: 1354993
zbMATH Keywords: \(\gamma\)-open set
zbMATH Keywords: preopen
zbMATH Keywords: semi-open
zbMATH Keywords: \(\alpha\)-open
zbMATH Keywords: \(\beta\)-open
zbMATH Keywords: locally closed
zbMATH Keywords: regular open set
reviewed by: Bruce R. Wenner
MaRDI profile type: Publication
Recommended article: Q4287220
Recommended article: Q4876165
Recommended article: On a topology between Tα and Tγα
Recommended article: Q5748091
Recommended article: Further remarks on the formula for \(\gamma\)-interior
Recommended article: Q4843494
Recommended article: Generalized open sets in generalized topologies
Recommended article: Q4789999
Recommended article: More properties of generalized open sets in generalized topological spaces
Recommended article: Q3172551
description: scientific article; zbMATH DE number 1354993