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:

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