## Spheres

The unit sphere in any $d$–dimensional space $\mathbb{R}^d$ is defined to be the set $\mathbb{S}_{d-1} = \big\{ (x_1,x_2,\dotsc,x_d) \in \mathbb{R}^d : \sum_{k=1}^d x_k^2=1 \big\}.$  The 1-dimensional sphere is, of course, the circle, that we can parametrize by angles:

$\varphi \colon [0,2\pi) \in \theta \mapsto e^{i\theta} = \cos \theta + i \sin \theta \equiv (\cos \theta, \sin \theta) \in \mathbb{S}_1$

While understanding the definition of homeomorphism, we worked on such a map from a disk $D_2$ to a square $\square_2.$  In this section we are going to use this idea to establish a homeomorphism between the 2-dimensional sphere $\mathbb{S}_2 \subset \mathbb{R}^3$, and the quotient space formed by a proper identification of points in the border of a square $\square_2 \subset \mathbb{R}^2.$  The construction goes as follows:

1. Scale the unit sphere by a half, and shift it vertically up by half unit, so the center is at $(0,0,\frac{1}{2}).$
2. The plane $\{ x_3=0\}$ intersects this sphere at the origin.  We identify this point with the center of the disk $D_2.$
3. For each $0 \leq \lambda < 1,$ the horizontal plane $\{x_3=\lambda\}$ intersects this sphere in the circle $\big\{ (x_1,x_2,\lambda) \in \mathbb{R}^3 : x_1^2 + x_2^2 + (\lambda - \frac{1}{2})^2 = \frac{1}{4} \big\}:$ the center is at $(0,0,\lambda),$ and the radius is $\sqrt{\frac{1}{4}-(\lambda-\frac{1}{2})^2}.$
We consider a map that scales this circle to that of radius $\lambda,$ and “places it” on the unit disk $D_2$ in a natural way.  So far, the map so constructed is injective, onto and continuous.
4. The issue now is what to do with the last non-void intersection of the sphere with horizontal planes.  This happens at the horizontal plane $\{ x_3 = 1\},$ and the result is the only point $(0,0,1).$  We overcome this situation by mapping the whole point into the last circle in $D_2$, and identifying the whole circle.  We accomplish this by creating the equivalence relation $(x_1,x_2) \sim (y_1,y_2)$ defined by the following rules:
1. $x_1=y_1$ and $x_2=y_2,$ or
2. $x_1^2+x_2^2=y_1^2+y_2^2=1.$

We have thus created a homeomorphism $\varphi_1 \colon \mathbb{S}_2 \to D_2.$  Compose this with an homeomorphism $\varphi \colon D_2 \to \square_2,$ and notice that the equivalence relation defined above turns into the equivalence relation $(x_1,x_2) \sim (y_1,y_2)$ defined by the following rules:

1. $x_1=y_1$ and $x_2=y_2$, or
2. $\mathop{\text{max}}\big(\lvert x_1\rvert, \lvert x_2\rvert \big) = \mathop{\text{max}}\big( \lvert y_1\rvert, \lvert y_2\rvert \big) = 1.$

Graphically, we represent this set with the diagram below: