I have been given the following definition:
Given a equivalence relation $R$ on a set $X$, a quotient of $X$ modulo $R$ is a pair $(Y, \pi$) consisting of a set $Y$ and a surjection $\pi : X \rightarrow Y$ (called the quotient map) with the property that $\pi(x) = \pi(y) \Leftrightarrow (x, y) \in R$.
Now the question is the following:
Describe the equivalence relation $R$ on $[0, 1]\times[0,1]$ that encodes the gluing to construct a torus. Then prove that the explicit model given by
$$T = \bigg\{(x, y, z) \in \mathbb{R}^{3} : \Big(\sqrt{x^2 + y^2} - R\Big)^2 + z^2 = r^2\bigg\}$$
is a quotient of $[0,1]\times[0,1]$ modulo $R$ in the sense of the previous definition (of course you also have to describe the map $\pi$).
So my equivalence relation:
$(x, y)\sim(w, z)\Leftrightarrow (x = 0, w = 1, y = z), (x = 1, w = 0, y = z), (y = 0, z = 1, x = w), (y = 1, z = 0, x = w)$.
I can't solve the rest of the problem. I lack understanding of what $\pi$ exactly is and how to relate $\pi$ to the explicit model for the torus. And what is $Y$ in this case? Some help with these concepts would be much appreciated!
Hint: Parametrize the torus with two families of orthogonal circles in the natural way. Then consider $f:[0,1] \times [0,1] \to S_1 \times S_1$ given by $(u,v) \mapsto (\cos 2\pi u, \sin 2\pi u,\cos 2\pi v, \sin 2\pi v)$ and $R$ given by $(u,v) R (u',v') \iff f(u,v) = f(u',v')$. Make sure you see that $R$ identifies opposite sides of the square.