On function spaces topologies in the setting of Čech closure spaces
type: Item
instance of: scholarly article
title: On function spaces topologies in the setting of Čech closure spaces
zbMATH Open document ID: 1230.54020
author: Dimitrije Andrijević
author: Milena Jelić
author: Mila Mršević
published in: Topology and its Applications
publication date: 2011-08-04T00:00:00Z
review text: A set \(X\) is called a Čech closure space \((X, u)\) in case there is an operator \(u\) defined on all subsets \(A\subset X\) with \(u(A)\subset X\) such that {\parindent7.5mm \begin{itemize}\item[(i)] \(u(\emptyset)= \emptyset\); \item[(ii)] \(u(A)\supset A\); \item[(iii)] \(u(A\cup B)= u(A)\cup u(B)\). \end{itemize}} It is not required that \(u(u(A))= u(A)\). Sets of the form \(X\setminus u(A)\) are said to be open in \(X\). Let \((Y, v)\) be another closure space and \({\mathcal F}\) a collection of functions \(f\) from \((X, u)\) to \((Y, v)\). Separation axioms on \(X\) from \(T_0\) up to normal are defined in terms of \(u\), and those for \({\mathcal F}\) are defined in terms of the product closure operator on \(X\times Y\). Among the topologies on \({\mathcal F}\) studied in the present paper is the graph-open topology with basis \(\{f\in{\mathcal F}: G(f)\subset W\}\), where \(G(f)\) is the graph of \(f\) in \(X\times Y\) and \(W\) is open in \(X\times Y\). Various relations between separation properties in \(X\), \(Y\), and \({\mathcal F}\) are obtained. We say that \(f\) is continuous in case \(f(u(A))\subset v(f(A))\) for every \(A\subset X\). We denote by \({\mathcal C}(X, Y)\) the set of all continuous functions from \(X\) to \(Y\). The authors show, among other things, that if \((X, u)\) is regular, \(Y\) is a topological space, and \({\mathcal F}\subset{\mathcal C}(X, Y)\), then the compact-open topology on \({\mathcal F}\) is contained in the graph-open topology.
reviewed by: James V. Whittaker
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 54A05
Mathematics Subject Classification ID: 54D10
zbMATH DE Number: 5935886
zbMATH Keywords: Čech closure space
zbMATH Keywords: compact-open topology
zbMATH Keywords: compact-interior topology
zbMATH Keywords: graph-open topology
zbMATH Keywords: graph-interior topology
MaRDI profile type: Publication
full work available at URL: j.topol.2011.05.013
OpenAlex ID: W1963614438
cites work: Some properties of hyperspaces of Cech closure spaces with Vietoris-like topologies
cites work: Topologies for function spaces
cites work: Q5516652
cites work: Q3725164
cites work: Q3733752
cites work: Q4150804
cites work: On topologies for function spaces
cites work: Q4952037
cites work: Q4526529
cites work: Weakly continuous, weakly \(\vartheta\)-continuous, super-continuous and topologies on function spaces
cites work: Closure properties of function spaces
cites work: Q3749745
cites work: Proper and admissible topologies in closure spaces
cites work: On \(\theta\)-connectedness and \(\theta\)-closure spaces
cites work: Q4064159
cites work: Graph Topology for Function Spaces
cites work: Q5647612
DOI: 10.1016/J.TOPOL.2011.05.013
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description: scientific article; zbMATH DE number 5935886