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Let $X,Y$ be topological spaces, let $Map(X,Y)$ be the spaces of all continuous maps from $X$ to $Y$, equipped with compact-open topology.

Is the pairing $X\times Map(X,Y)\rightarrow Y, \ (x,\phi)\mapsto \phi(x)$ continuous? If it is, how to show this?

THANKS A LOT!

citadel
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Zoudelong
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  • If you take $Y = [0, 1]$ and $X$ be any Tychonoff space that isn't locally compact, then your map isn't continuous. – Jakobian Sep 17 '23 at 17:23
  • Thus, at least for Tychonoff spaces, we have that $(x, \varphi)\mapsto \varphi(x)$ is continuous iff $X$ is locally compact. – Jakobian Sep 17 '23 at 17:41
  • You should consult a textbook on general topology (function spaces, evaluation map). – Paul Frost Sep 18 '23 at 10:10

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