Submanifold of dimension k in $\mathbb{R}^n$ If $\Sigma \subset \mathbb{R}^n$ is a submanifold of dimension k, Why can't we just say it's a subset of $\mathbb{R}^k$ or is that usually implied, or does n have to be atleast k+1?

The sphere $S^2$ is a submanifold of $\mathbb{R}^3$ of dimension 2, but it is not a subset of $\mathbb{R}^2$.
