 Solution

A subset $Z$ of an $m$-dimensional smooth manifold $M$ is an $r$-dimensional embedded submanifold if, for each $p\in Z$, some coordinate chart $x:U\to V\subset {\mathbb{R}}^m$ containing $p$ satisfiesThe induced topology on $Z$ is the subspace topology, and the first $r$ coordinates restricted to $Z$ give its smooth atlas. To see their smooth transition maps agree, restrict the ambient smooth transitions to the coordinate planes; their inverses restrict in the same way. The inclusion then has injective differential of a smooth map and is a homeomorphism onto its image. Conversely this is the local coordinate description of a smooth smooth embedding. An embedded submanifold need not be a closed subset of $M$.