\((\delta, \delta ')\)-continuity on generalized topological spaces
type: Item
instance of: scholarly article
title: \((\delta, \delta ')\)-continuity on generalized topological spaces
zbMATH Open document ID: 1265.54017
published in: Acta Mathematica Hungarica
publication date: 2011-02-25T00:00:00Z
review text: W. K. Min studies continuity for generalised topological spaces where the empty set is open and the arbitrary union of open sets is again open; intersections of open sets need not to be open. The resulting interior-operator \(i(A)\) being the union of all open subsets of \(A\) satisfies that \(i(A) \subseteq A\), \(A \subseteq B \Rightarrow i(A) \subseteq i(B)\) and \(i(i(A)) = A\). Hence the closure-operator in such a generalised topology is just any closure operator satisfying the usual axioms and one can therefore identify the generalised topological spaces with the spaces of such closure operators. Given an open topology \(\mu\) of this type, W. K. Min derives another topology \(\delta\) which contains all those subsets \(A\) of \(X\) such that for each \(x \in A\) there is a \(\mu\)-closed set \(B \subseteq A\) with \(x\) being in the \(\mu\)-interior of \(B\). Given now two generalised topological spaces \((X,\mu)\) and \((Y,\mu')\), the author studies when a function \(f: X \rightarrow Y\) is \((\delta,\delta')\)-continuous with respect to \(\delta,\delta'\) derived from \(\mu,\mu'\). The notion of \((\delta,\delta')\)-continuity is independent of \((\mu,\mu')\)-continuity, but both types of continuities imply that the function is almost \((\mu,\mu')\)-continuous and weakly \((\mu,\mu')\)-continuous, where these notions were introduced by the author in prior work. The notions of \((\mu,\mu')\)-continuity, \((\delta,\delta')\)-continuity and almost \((\mu,\mu')\)-continuity coincide on a suitable generalisation of the notion of regular spaces to this form of generalised topology.
reviewed by: Frank Stephan
Mathematics Subject Classification ID: 54A05
Mathematics Subject Classification ID: 54C08
zbMATH DE Number: 5857801
zbMATH Keywords: \((\mu,\mu')\)-continuous
zbMATH Keywords: almost \((\mu,\mu')\)-continuous
zbMATH Keywords: weakly \((\mu,\mu')\)-continuous
zbMATH Keywords: \((\delta, \delta ')\)-continuous
zbMATH Keywords: \(\delta\)-open
zbMATH Keywords: G-regular
author: Won Keun Min
MaRDI profile type: Publication
full work available at URL: s10474-010-0036-4
OpenAlex ID: W1990725373
cites work: Generalized topology, generalized continuity
cites work: Generalized open sets in generalized topologies
cites work: \(\delta\)- and \(\theta\)-modifications of generalized topologies
cites work: Some results on generalized topological spaces and generalized systems
cites work: Weak continuity on generalized topological spaces
cites work: Almost continuity on generalized topological spaces
DOI: 10.1007/S10474-010-0036-4
description: scientific article; zbMATH DE number 5857801