$M \subseteq R^d $ is a compact submanifold of $R^d$ and contains at least 2 points. Show that $M$ has no global parametrization.
How can I show this?
Compact submanifold has no global parametrization
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real-analysis
manifolds
1 Answers
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If it were the case, then the chart would be a homeomorphism, but the only open/compact sets in $\mathbb R^d$ are the empty set or for $d=0$ the set $\{0\}$, both in contradiction with the requirement that the submanifold has at least two points.