Necessary and sufficient conditions for admissible set open topologies (Q1731345)

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.