Let $M$ be a (topological) $m$-manifold and $N \subset M$ be an embedded $n$-manifold. If we assume that $N$ is locally flat, that is, for all $x\in N$, there is a neighborhood $U\subset M$ of $x$ such that the topological pair $(U,U\cap N)$ is homeomorphic to the pair $(\mathbb {R} ^{n},\mathbb {R} ^{d})$, with the standard inclusion of ${\displaystyle \mathbb {R} ^{d}\to \mathbb {R} ^{n}}$.
It is easy to show that $n \leq m$. How can this result be shown if $N$ is not locally flat?
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.