# Definition
> **The 2-sphere $S^2$**
> The set of points in $\pmb{x} \in \mathbb{R}^3$ such that $\{\pmb{x} \in \mathbb{R}^3 \,:\, |x| = r\}$ where $r \in \mathbb{R}$ is a constant known as the radius of the sphere. It is said to be *[[Embedding|embedded]]* in $\mathbb{R}^3$.
# Stereographic Projection
An example of a map for the sphere is the so-called [[stereographic projection]], demonstrated in the figure below. Note that this map only fails at a single point $N$, thus, we need at least two charts to cover the whole sphere.
![[Pasted image 20210123183150.png]]
The 2-sphere can be written as as [[quotient space]]:
$
S^2 = D^2/S^1
$
where $S^1$ is the [[circle|circle]] and $D^2$ is a 2-dimensional disc.