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Let $X,Y$ be topological spaces, $C(X,Y)$ the set of continuous maps from $X$ to $Y$ with some topology $\mathcal{T}$. I'm trying to show that if the evaluation map $\operatorname{ev}:(f,x)\mapsto f(x)$ is continuous, then $\mathcal{T}$ contains the compact-open topology, with a subbasis given by sets of the form$$S(K,U):=\{f\in C(X,Y)\,|\,f(K)\subseteq U\}$$for $K\subseteq X$ compact and $U\subseteq Y$ open.

Here's what I've done so far:

Given some $f_0\in S(K,U)$, I'm trying to find an open neighbourhood of that point in $S(K,U)$. $\operatorname{ev}^{-1}(U)$ is given by$$\operatorname{ev}^{-1}(U)=\{(f,x)\in C(X,Y)\times X\,|\,f(x)\in U\}$$Assuming $S(K,U)$ is not empty, take some $f_0\in S(K,U)$ and some $x_0\in K$, clearly $(f_0,x_0)\in \operatorname{ev}^{-1}(U)$. Since $\operatorname{ev}^{-1}(U)$ is open in the product topology, there exist sets $V\subseteq C(X,Y)$ and $W\subseteq X$ open, such that $$V\times W\subseteq\operatorname{ev}^{-1}(U)$$If $K\subseteq W$, then $V\subseteq S(K,U)$, since by the condition above all $f\in W$ must map $W$ to $U$, so they must also map $K$ to $U$. So I would like to always find some open $W\subseteq X$ containing $K$ which has a partner $V$ such that $V\times W\subseteq\operatorname{ev}^{-1}(U)$ as above. I've tried taking $W=f_0^{-1}(U)$ which must contain $K$, but that has not seemed to help.

Any help is appreciated.

John Doe
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1 Answers1

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Let, $f\in S(K,U)$ where $K\subseteq X$ and $U\subseteq Y$ are compact and open respectively. Now the evaluation map $ev:C(X,Y)\times X\to Y$ is continuous. So for each $x\in K$ we have, $ev(f,x)\in U$ so by local continuity there exists an open set $V_x$ in $C(X,Y)$ containing $f$ and an open set $U_x$ in X containing $x$ such that $g\in V_x$ and $y\in U_x\implies g(y)\in U$. Now, $\{U_x\,:\,x\in K\}$ is an open cover for $K$. Since, $K$ is compact, it has a finite subcover $\{U_{x_i}\,:\,1\leq i\leq n\}$. Now set, $V=\cap_{i=1}^{n}\,V_{x_i}$ then, $V$ an open set in $C(X,Y)$ containing $f$. Show that, $V\subseteq S(K,U)$.

Utsabraj Sarkar
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