1

My book is An Introduction to Manifolds by Loring W. Tu.

Corollary 8.7 is (smooth) invariance of dimension, and Theorem 22.3 is smooth invariance of domain.

I view these as converses and think of combining these two together to say

  • Let $U$ be an open subset of $\mathbb R^n$. Let $S$ be a subset of $\mathbb R^m$. Let $U$ be diffeomorphic to $S$. Then $m=n$ if and only if $S$ is open in $\mathbb R^m$. $\tag{1}$

By generalizing Corollary 8.7 based on Corollary 8.6 to get

  • Let $U$ be an open subset of a smooth $n$-manifold $N$. Let $S$ be a subset of a smooth $m$-manifold $M$. Let $U$ be diffeomorphic to $S$. Then $m=n$ if $S$ is open in $M$. $\tag{2}$

I think I can generalize $(1)$ based on Remark 22.5 to get

  • Let $U$ be an open subset of a smooth $n$-manifold $N$. Let $S$ be a subset of a smooth $m$-manifold $M$. Let $U$ be diffeomorphic to $S$. Then $m=n$ if and only if $S$ is open in $M$. $\tag{3}$

Are these correct?

  • @freakish Thanks. What do you mean? I think they are incorrect if they said "and" instead of "if and only if". Take (1) for example. How do you know $m=n$ without assuming $S$ is open in $\mathbb R^m$? Or if you prove $m=n$ after proving $S$ is open in $\mathbb R^m$, then how do you prove $S$ is open in $\mathbb R^m$ without assuming $m=n$? –  Jul 20 '19 at 11:30
  • 1
    Sorry, I misread the whole thing. Yeah, you are right, the formulation is fine. – freakish Jul 20 '19 at 11:48
  • @freakish I really don't think those hold with "and" instead of "if and only if"...oh ok you deleted all your comments. Thanks for the effort and the reread! You can answer if you want. –  Jul 20 '19 at 11:48

0 Answers0