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Let $M$ be a topological manifold, and consider $N\subset M$ with the subspace topology.

Assume $N$ is a topological manifold of codimension $0$. Must it be open in $M$?

It is clearly the case for any locally flat submanifold. But does it hold in general? (See discussion on possibilities of defining submanifolds).

(I know this holds in the smooth case, but there we have the inverse function theorem, so we get a local diffeomorphism).

Community
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Asaf Shachar
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1 Answers1

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It is a direct consequence of the invariance of domain.

PseudoNeo
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    How? I am afraid I do not see it yet... – Asaf Shachar Sep 07 '15 at 16:44
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    The inclusion $i : N \to M$ is a continuous one-to-one map between two $\dim M$-dimensional manifolds, so by the invariance of domain it is open. $N$ is tautologically open in itself, so $ N = i[N]$ is open in $M$. – PseudoNeo Sep 07 '15 at 16:51