The Real Projective Plane

Consider in the sphere $\mathbb{S}_2 \subset \mathbb{R}^3,$ the relation induced by identification of antipodal points; that is, given $z_1, z_2 \in \mathbb{S}_2$ , set $z_1 \sim z_2$ if and only $z_1 = \pm z_1.$ The corresponding quotient space $\mathbb{P} = \big( \mathbb{S}_2 / \sim \big)$ is what we call real projective space.

Since we are interested in the topological properties of this space, we actually define a real projective space to be any homeomorphic set to $\mathbb{P}.$ Among those, we are interested in one that can be realized from a square, by identification of its sides (in a similar manner as we did with the torus). We proceed as follows:

Assume that we start from the unit sphere $\{ (x_1,x_2,x_3) \in \mathbb{R}^3 : x_1^2+x_2^2+x_3^2=1\}$ , and note that the upper hemisphere $H=\{ (x_1,x_2,x_3) \in \mathbb{R}^3 : x_1^2+x_2^2+x_3^2=1, x_3 \geq 0 \}$ contains at least one of each pair of antipodal points. If both antipodal points occur in $H$ , they will necessarily lie over the circle $\{ (x_1, x_2, 0) \in \mathbb{R}^3 : x_1^2+x_2^2=1\}.$ The hemisphere $H$ is obviously homeomorphic to a disk (by a simple vertical projection onto the plane $\{x_3=0\},$ for example). And the disk is homeomorphic to a square, so we may use a composition of both to realize a homomorphism from $H$ to $\square_2.$

Define in $H$ an equivalence relation that identifies two antipodal points on the border, and notice that the homeomorphism just computed takes that identification to the following: Given $(x_1,x_2), (y_1,y_2) \in \square_2$ , it is $(x_1,x_2) \sim (y_,y_2)$ if

$x_1=y_1$ and $x_2=y_2$ , or
$x_1y_1 = -1$ and $x_2=1-y_2$ , or
$x_2y_2=-1$ and $x_1=1-y_1$ .

A diagram representing the quotient space $\big( \square_2 / \sim \big)$ is presented below:

The punch-line is, of course, to construct a homeomorphism from the real projective plane $\mathbb{P}$ as defined above, to the quotient space $\big( \square_2 / \sim \big).$ The reader should not have much trouble to give an analytic expression of such a map following the steps above.

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Francisco Blanco-Silva