= 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 $p$ satisfies
$$
x(U\cap Z)=V\cap(\mathbb R^r\times\{0\}^{m-r}).
$$
The 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 $M$.
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