Construct a diffeomorphism $\psi: B_1 \to \epsilon\text{-neighborhood of } K$, where $K$ is a subset of a smooth manifold.

Question

I'm currently working through a paper by Brezis on the topology of Sobolev spaces. Right now I am having trouble understanding the following note made by Brezis.

Let $M$ be a compact and smooth manifold and $K \subset M$. Then there exists a diffeomorphism $$\psi: B_1(0) \to \epsilon\text{-neighbourhood of } K$$ (where $B_1(0) \subset \mathbb{R}^n$ and $\dim M = n$).

Does it have to do with $K_\epsilon$ being an open subset of the smooth manifold $M$ and thus a smooth manifold itself? Does that already imply the existence of such diffeomorphism? I am a little confused right now.

I'd appreciate any help.

Answer 1

The author is not claiming that such a diffeomorphism always exists. For instance, if $K = M = S^1$ or $M = S^1\times S^1, K = S^1 \times \{0\}$, then it is clear that no neighborhood of $K$ can be homeomorphic to an open ball.