Equivalence relation on $S^2$ and unit square make same space

Question

I think I understand a bit of this task, but I hope someone can look critical to my answers:

Let $X$ be the space obtained from the sphere $S^2$ by gluing the north and the south pole (with the quotient topology). Show that $X$ can be obtained from a square $[0,1]\times[0,1]$ by glueing some of the points on the boundary (note: you're not allowed to glue a point in the interior of the square to any other point). More precisely:

1. Describe the equivalence relation $R_0$ on $S^2$ encoding the glueing that defines $X$.
2. Make a picture of $X$ in $\mathbb{R}^3$.
3. Describe an equivalence relation $R$ on $[0,1]\times[0,1]$ encoding a glueing with the required properties
4. Show that, indeed, $X$ is homeomorphic to $[0,1]\times[0,1]/R$ (provide as many arguments as you can, but do not write down explicit maps- instead, indicate them on the picture).

My attempt:

1. Describe the equivalence relation $R_0$ on $S^2$ encoding the glueing that defines $X$.

I thought $R_0=\{(x,y)\in X\times X: [x=(0,0,1) \text{ and } y=(0,0,-1)] \text{ or } [x=(0,0,-1) \text{ and } y=(0,0,1)] \text{ or } [x=y]\}$. With this, $R_0$ defines an equivalencerelation, because it satisfies the axioms.

2. Make a picture of $X$ in $\mathbb{R}^3$.

I will describe my picture (it is not so pritty). I thought of it as a torus, but then without a hole in the middle. This is just where the north and south poles are glued together, a little bit like a node.

3. Describe an equivalence relation $R$ on $[0,1]\times[0,1]$ encoding a glueing with the required properties.

I thought of it as glueing the unit square as a torus, and then glueing two opposite boundaries of the unit square to one single point, as you can see in the picture (not my best picture, I must confess)

4. Show that, indeed, $X$ is homeomorphic to $[0,1]\times[0,1]/R$ (provide as many arguments as you can, but do not write down explicit maps- instead, indicate them on the picture).

I actually don't know what to do anymore to prove they are homeomorphic, you can see it in the picture, but I think I still have something to prove. Who can help me?