Q:

Dimension of topological submanifold is at most that of the containing manifold

Let $M$ be a (topological) $m$-manifold and $N \subset M$ be an $n$-manifold. If we assume that $N$ is locally flat, then it is easy to show that $n \leq m$. How can this result be shown if $N$ is not locally flat?

A:

If we focus on the neighbourhood within $M$
of a point of $N$ we get an embedding $i$ of
$\Bbb R^n$ in $\Bbb R^m$. Compose this with the natural embedding
$j:\Bbb R^m\to\Bbb R^n$ (if $m<n$). Then $j\circ i:\Bbb R^n\to\Bbb R^n$
is an embedding, and by Brouwer's invariance of domain theorem, must have open image. But being contained in a  proper linear subspace, the image cannot be open,
and we get a contradiction.

