Consider a manifold smooth manifold $N$ smoothly embedded in another manifold $M$ of the same dimension. Is it true that $N$ is open in $M$? I think this is true, due to the open mapping theorem.
If so, is this also true with less regularity?
differential-geometrygeneral-topologymanifoldssmooth-manifolds
Consider a manifold smooth manifold $N$ smoothly embedded in another manifold $M$ of the same dimension. Is it true that $N$ is open in $M$? I think this is true, due to the open mapping theorem.
If so, is this also true with less regularity?
Best Answer
It is true for topological manifolds (without boundary). This is a consequence of the invariance of domain. See for example
Version of Invariance of Domain for n-manifolds
https://en.wikipedia.org/wiki/Invariance_of_domain
Given an embedding $f : N \to M$, each $x \in N$ has an open chart neighborhood $U$ which is mapped homeomorphically into an open chart neighborhood $V$ of $f(x)$. Both $U,V$ are homeomorphic to open subsets of $\mathbb{R}^n$, hence invariance of domain implies that $f(U)$ is open in $V$ and therefore open in $M$. Since $f(U) \subset f(N)$, we see that $f(N)$ is open in $M$.