Necessary and sufficient conditions for admissible set open topologies
type: Item
instance of: scholarly article
title: Necessary and sufficient conditions for admissible set open topologies
zbMATH Open document ID: 1415.54009
author: Abderrahmane Bouchair
author: Imane Dekkar
published in: Topology and its Applications
publication date: 2019-03-13T00:00:00Z
review text: Given topological spaces $X$ and $Y$, the expression $C(X,Y)$ stands for the set of all continuous functions from $X$ to $Y$. If $\alpha$ is a family of subsets of $X$, then the \textit{set-open topology} $\tau_\alpha$ on $C(X,Y)$ is generated by the family ${\mathcal S}= \{[A,V]: A\in \alpha$ and $V$ is an open subset of $Y\}$ as a subbase. Here $[A,V]=\{f\in C(X,Y): f(A)\subset V\}$ for any open subset $V$ of the space $Y$ and $A\in \alpha$. \par A topology $\tau$ on the set $C(X,Y)$ is called \textit{admissible} if the evaluation map $e:C(X,Y) \times X\to Y$ defined by $e(f,x)=f(x)$ for any $f\in C(X,Y)$ and $x\in X$ is continuous on the space $(C(X,Y), \tau)\times X$. A family $\alpha$ of subsets of $X$ is called \textit{regular} if for any $x\in X$ and any open subset $U$ of the space $X$ such that $x\in U$, there exists $A\in\alpha$ with $x\in \text{Int}(A)\subset A\subset U$. \par A topological space $Y$ is said to be \textit{equiconnected} if there exists a continuous function $\Psi:Y\times Y\times [0,1]\to Y$ such that $\Psi(x,y,0)=x$, $\Psi(x,y,1)=y$ and $\Psi(x,x,t)=x$ for all $x,y\in Y $ and $t\in [0,1]$. The main result of the paper states that for any equiconnected $T_1$-space $Y$, if $\alpha$ is a family of subsets of a completely regular space $X$, then the set-open topology $\tau_\alpha$ on $C(X,Y)$ is admissible if and only if $\alpha$ is a regular family.
Mathematics Subject Classification ID: 54C35
zbMATH DE Number: 7035719
zbMATH Keywords: function space
zbMATH Keywords: set-open topology
zbMATH Keywords: admissible topology
zbMATH Keywords: \(M\)-space
zbMATH Keywords: \(M\)-flow
reviewed by: Vladimir Tkachuk
MaRDI profile type: Publication
full work available at URL: j.topol.2019.02.008
OpenAlex ID: W2912810423
cites work: Q4261832
cites work: A topology for spaces of transformations
cites work: Topologies for function spaces
cites work: Essentiality and injectivity
cites work: Comparison of some set open topologies on \(C(X, Y)\)
cites work: Injective topological fibre spaces
cites work: Injectivity and sections
cites work: Q4150804
cites work: On fibre spaces. I
cites work: Admissible topologies on \(C (Y, Z)\) and \({\mathcal O}_Z (Y)\)
cites work: On the greatest splitting topology
cites work: On dual topologies
cites work: Q5156809
cites work: Q5389909
cites work: Monoids, acts and categories. With applications to wreath products and graphs. A handbook for students and researchers
cites work: Topological properties of spaces of continuous functions
cites work: Some properties of set-open topologies
cites work: Q3052959
cites work: Q3597355
DOI: 10.1016/J.TOPOL.2019.02.008
description: scientific article; zbMATH DE number 7035719