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Let $X,Y,Z$ be Hausdorff spaces and suppose that $Z\subset X$. Endow $C(X,Y)$ and $C(Z,Y)$ with the compact-open topologies and define the map $\rho$ as \begin{align} \rho:&C(X,Y)\rightarrow C(Z,Y)\\ &f\mapsto f|_Z. \end{align}

Is the map $\rho$ continuous?

I see this "type of" operation used all the time in Sheaf theory but I never stopped to wonder if it was indeed continuous?

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    $\begingroup$ If I am not mistaken it follows immediately from the definition of the compact-open topology together with the fact that the image of a compact $K \subset Z$ under the continuous map $Z \to X$ is compact. $\endgroup$
    – Reid Barton
    Commented May 23, 2019 at 23:25
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    $\begingroup$ If you like, I'll accept that if you post it as an answer. $\endgroup$
    – ABIM
    Commented May 24, 2019 at 13:24

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If $K$ is compact in $Z$ and $U$ open in $Y$, then $K$ is still compact as a subset of $X$ as well (compactness is absolute, or maybe use that $i[K]$ is compact where $i: Z \to X$ is the continuous canonical embedding..)

$\rho^{-1}[[K,U]]=\{f \in C(X,Y)\mid (f\restriction_Z)[K] \subseteq U\} = \{f \in C(X,Y)\mid f[K] \subseteq U\} =[K,U]$ which is open in $C(X,Y)$ by definition. As the sets $[K,U]$ form a subbase for $C(Z,Y)$, $\rho$ is continuous.

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  • $\begingroup$ Thanks Henno :) $\endgroup$
    – ABIM
    Commented May 27, 2019 at 23:03

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