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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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1 Answers 1

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