Smooth embedding that isn't an open or closed map

Question

In Lee's book there is an exercise:

Give an example of a smooth embedding that is neither an open map nor a closed map.

I'm confused; a smooth embedding is a homeomorphism which is both an open and closed map.

Am I missing something here?

Answer 1

Take $f:\mathbb{R} \to \mathbb{R}^2,$ given by $f=\iota_{S^1 \backslash N}^{\mathbb{R}^2} \circ \mathrm{steo}^{-1},$ where given $A \subset B$, $\iota_A^B$ is the inclusion mapping, and $\mathrm{steo}$ is the stereographic projection from the north pole $N$.

Answer 2

Let $J$ denote the open interval $(0,1)$.

Let $f:J \to \mathbb{R^2}$ be defined by $f(x) = (x,0)$.

Then $f$ is a smooth embedding but $f$ is neither open nor closed.

For example, $J$ is both open and closed in $J$, but $f(J)$ is neither open nor closed in $\mathbb{R}^2$.