Is this statement about manifold true?

Question

Suppose $M$ is a closed $k-$manifold in $\mathbb R^n$ without boundary, can we always find a smooth function $f:\mathbb R^n\to\mathbb R^{n-k}$ such that $M$ is the level set where $f=0$?

Answer 1

In fact, much more is true. Every closed subset of $\Bbb R^n$ is the zero set of some function $\Bbb R^n \rightarrow \Bbb R$. See Every closed subset $E\subseteq \mathbb{R}^n$ is the zero point set of a smooth function