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:

\begin{equation} \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 \end{equation}

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: 