TITLE: An injective immersion that is not a topological embedding
QUESTION [17 upvotes]: On page 86 of John Lee's Introduction to smooth manifolds there is an example of an injective immersion that is not a topological embedding:
$\beta : (-\pi, \pi) \to \mathbb{R}^2$, defined by $\beta(t) = (\sin{2t}, \sin{t})$, or pictorially:

It is explained that, although $\beta$ is an injective immersion, it is not a smooth embedding since the image is compact while the domain is not.
My understanding is that the image, while bounded in $\mathbb{R}^2$, is an open subset of the plane, whereas the statement claims that it is not.
Would anyone please explain why the image is compact? Thank you.

REPLY [2 votes]: Note that interior of $\beta ((-\pi,\pi)) \subset \mathbb{R}^2$ is empty. So it cannot be an open subset of $\mathbb{R}^2$. Now take any open cover $\{U_i\}_{i \in I}$ of the image. For some $j \in I$, we must have  $0 \in U_j$. Clearly, the rest of the image can be covered by finitely many $U_k$'s where $k \in I$. So we started with any open cover and find a finite subcover. That is, image is compact subset of $\mathbb{R}^2$.