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. $[0,1)\subset[0,1].$
2026-04-24 12:17:56.1777033076
If $X$ is a non-compact metric space, can $X^n$ ever be compact?
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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.