The torus $\mathbb{T}^2$ is usually defined as the quotient space of the unit square formed by identifying the top and bottom halves of the unit square, and then the left and right halves of the unit square.
Hence is the torus $\mathbb{T}^2$ defined as the quotient space $$\mathbb{T}^2 = I \times I / \sim_{tb} / \sim_{lr}$$
where $I = [0, 1] \subseteq \mathbb{R}$ and $\sim_{tb}$ is the equivalence relation $(x, 1) \sim_{tb} (x, 0)$, and $\sim_{lr}$ is the equivalence relation $(0, y) \sim_{lr} (1, y)$?
Best Answer
That is correct - and the two-step definition (quotient space by one relation, and then quotient space of the first quotient space by the second relation) has a simple and intuitive geometric/topological interpretation: When you factor by the first relation, what you get is a (hollow) cylinder of finite length. The second relation identifies one "edge" of the cylinder with the other, to form the torus.