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Do there exist metric spaces $X$ such that $X^n$ is compact even though $X$ is not? Since compact spaces can have non-compact subspaces, e.g. $[0,1)\subset[0,1].$

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    $\begingroup$ If $A \times B$ is compact (for the product topology), then both $A$ and $B$ are compact. $\endgroup$ – Watson May 12 '16 at 11:20
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    $\begingroup$ I don't even know why I wrote that. $\endgroup$ – user339043 May 12 '16 at 11:20
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    $\begingroup$ Yes, if $n=0$... $\endgroup$ – Johannes Huisman May 12 '16 at 14:58
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No, since the image of a compact map by a continuous map is compact and the projection $p:X^n\rightarrow X$ is continuous and surjective.

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