Let $M$ be a compact submanifold of $\mathbb R^N$, is it true that $M\times \mathbb R$ is a compact submanifold of $\mathbb R^{N+1}$?
product of a compact and non compact submanifolds
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general-topology
manifolds
1 Answers
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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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2@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. – 2012-09-26