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.