RP2 is homeomorphic to $(I \times I)/ {\sim}$

Question

the problem says

Show tha $\mathbb{RP}^2$ is homeomorphic to $(I \times I)/ {\sim}$, where $(s, 0) \sim (1 − s, 1)$ and $(0, t) \sim (1, 1 − t)$, for all $s, t \in I , I = [0,1]$

I not know how prove this. How to use the equivalence relation?

Answer 1

$\mathbb{RP}^2$ is the quotient of $\mathbb{R}^3\setminus\{0\}$ by the equivalence relation $x\sim \lambda x$. We can do this quotient in two steps: First the equivalence relation $x\sim \lambda x$ with $\lambda>0$ and then identify $x$ with $-x$.

The first collapse gives us the sphere. And the last is identifying the antipodals.

Before we do that we can cut the sphere in two hemispheres. Flatten and turn them turn each into squares in your mind using your favorite homeomorphism between the disc and the square. Then flip one and identify them together according to $x\sim -x$, except for the edges.

The identification of the edges that remains to be done is the same $(I\times I)/\sim$ that you were given.

Answer 2

Hint: If $\mathbf{RP}^2$ is defined to be the space of lines passing through the origin in $\mathbb{R}^3$, we need a continuous bijection between this space and the one given by that equivalence class.

Consider the space given by the equivalence class to be a quotient topology generated by this relation. Then try to find a suitable map between the two spaces.