 Solution

A $k$-dimensional embedded submanifold of ${\mathbb{R}}^n$ is a subset $M$ such that near each $p\in M$ there is a smooth diffeomorphism of ambient open neighbourhoods that sends $M$ to the coordinate plane ${\mathbb{R}}^k\times \{0{\}}^{n-k}$. More precisely, for some open $U\ni p$ and diffeomorphism $\Phi :U\to V\subseteq {\mathbb{R}}^n$,Thus locally the subset has $k$ freely varying smooth coordinates.