Torus

Example

Let $\sim$ be the equivalence relation on the square $[0, 1] \times [0, 1]$ generated by $(0, y) \sim (1, y)$ and $(x, 0) \sim (x, 1)$ for all $x, y \in [0, 1]$. The quotient space $[0, 1]^2 / {\sim}$, with the quotient topology, is the torus $T^2$.

Remarks

The torus is homeomorphic to the product $S^1 \times S^1$ of two circles, with the product topology, and to $\mathbb{R}^2 / \mathbb{Z}^2$. It is also homeomorphic to a surface in $\mathbb{R}^3$: turn a circle about an axis in its plane that does not meet the circle. The torus is a compact, connected surface. It is not homeomorphic to the sphere.

Explanations

L1 human

Take a square sheet. Glue the left edge to the right edge to get a tube, and then glue the two end circles of the tube. The result is the surface of a doughnut.