Help to understand the claim with the stereographic projection

Question

I don't understand the following claim: We consider $\Bbb S^2$ as a subspace of $\Bbb R^3$. We have the stereographic projection $\rho: \Bbb S^2 \setminus \{(0,0,1)\} \to \Bbb R^2$, $\rho (x,y,z)=\frac{(x,y)}{1-z}$. Now $\rho$ is a homeomorphismus. Now they claim is that the set $\Bbb S^2 \setminus \{(0,0,1)\}$ is an open set (with the relativ topology??) containing all but one point on the sphere and so every point except for possibly $(0,0,1)$ has a euclidiean neighborhood. How they conclude this? Why is there a euclidean neighborhood? Can someone please helps me?

Answer 1

It's a bijection, which projects every point of the sphere at some point at a plane. Additionally, a stereographic projection makes circles from circles.

So possibly they mean a topology of a plane, projected onto a sphere?

PS
Possibly, in $\rho: \Bbb S^2 \setminus \{0,0,1\} \to \Bbb R^2$ you should use $\{(0,0,1)\}$ instead of $ \{0,0,1\}$ ...?

Answer 2

The conecpt of a "Euclidean neighborhood" occurs in the context of toplogical manifolds. See for example https://en.wikipedia.org/wiki/Topological_manifold. Quote from the section "Coordinate charts":

By definition, every point of a locally Euclidean space has a neighborhood homeomorphic to an open subset of $\mathbb R^n$. Such neighborhoods are called Euclidean neighborhoods.

More generally, we may say that an open neigborhood of a point $x$ of a topological space $X$ is a Euclidean neighborhood if it is homeomorphic to an open subset of some $\mathbb R^n$.

In your example $U = S^2 \setminus \{(0,0,1)\}$ is an open subspace of $S^2$ which is homeomorphic to $\mathbb R^2$. Thus each $x \in S$ has a Euclidean neighborhood (namely $U$).