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Let $M$ be a compact submanifold of $\mathbb R^N$, is it true that $M\times \mathbb R$ is a compact submanifold of $\mathbb R^{N+1}$?

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up vote 4 down vote accepted

It can’t be compact: for any $x\in M$, $\{x\}\times\Bbb R$ is a closed subset that isn’t compact.

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Unless $M$ is empty, of course. – Chris Eagle Sep 26 '12 at 10:20
@Chris: Well, the part after the colon is still true! :-) – Brian M. Scott Sep 26 '12 at 10:21
for each $x\in M$, $\{x\}\times \mathbb R$ is not compact because it is homeomorphic to $\mathbb R$ which is not compact. It remains to see why this implies that $M\times \mathbb R$ is not compact? – palio Sep 26 '12 at 10:27
@palio: Every closed subset of a compact space is compact, so if $M\times\Bbb R$ were compact, $\{x\}\times\Bbb R$ would be compact. – Brian M. Scott Sep 26 '12 at 10:30

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