Immersion of sphere into $\mathbb{R}^2$ This is the first time I ask a question, and as a French I will probably make a few mistakes. Sorry for that.
I want to show that there are no immersions between sphere and plane (in particular, I want to show that there is no immersion between $S^2 = \{ x^2+y^2+z^2=1 \}$ and $\mathbb{R}^2$).
I read somewhere online this very concise proof (from what looked like a trustable source, in a good university pdf course), but I don't understand it.
"Let $f$ be an immersion from $S^2$ to $\mathbb{R}^2$. The image of $f$ is open (since an immersion between two spaces of same dimension is an open map) and closed (by compactness). Thus this is $\mathbb{R}^2$ (by connectedness), which is absurd, by compactness."
Can you help me figure it out? Especially the first part : how can the argument of $f$ being an open map can be used to show that the image is open since $S^2$ is closed?

$S^2$ is indeed a closed subset of $\mathbb R^3$. However, when you consider an immersion $f: S^2 \mapsto \mathbb R^2$, you're dealing with $S^2$ as a topological space on its own. Therefore as for all topological space $X$, $X$ itself is open. That is an axiom of a topological space.
