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Suppose that $X$ and $Y$ are topological spaces, and we consider $Y^X$, i.e. the space of maps from $X$ to $Y$ with the compact-open topology. If $X$ is compact then can we say anything about $Y^X$?

Thanks! Jon

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Generalizations of Arzelà-Ascoli at Wikipedia – kahen Oct 13 '11 at 17:08
Is there any added assumption on $Y$? Is it $T_1$, or regular, etc. – Asaf Karagila Oct 13 '11 at 17:09
If $X$ is the one pointed space (compact), then $Top(X,Y) \cong Y.$ – jspecter Oct 13 '11 at 17:11

Suppose $X$ is a one-point space. Then $Y^X$ is naturally homeomorphic to $Y$, so is not compact unless $Y$ is.

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