Proof of 2-sphere being connected

Question

Is there a short proof for the 2-sphere being connected? I only saw proof for the n-sphere but that is more complex than I need it to be.

Answer 1

Stereographic projection from the north pole/south pole is a homeomorphism of a subset of $S^2$ with $\mathbb{R}^2$, hence that subset is connected. The sphere is the union of these two connected subsets, and the subsets have nonempty intersection, hence the $2$-sphere is connected.

Answer 2

Let $\mathbb{S}^2$ be the $2$-sphere. One has the following homeomorphism: $$\mathrm{St}_N:\left\{\begin{array}{ccc}\mathbb{S}^2\setminus\{(0,0,1)\}&\rightarrow&\mathbb{R}^2\\(x,y,z)&\mapsto&\displaystyle\left(\frac{x}{1-z},\frac{y}{1-z}\right)\end{array}\right..$$