---
tag: 03NM7DZXXC
kind: example
name: Torus
msc: [57K20]
instance_of: [0CQHFPGJ45]
uses: [0V8046R0P9, 0Z57EC6YX3, 0WH1GVBJNH]
sources:
  - url: https://en.wikipedia.org/wiki/Torus
    locator: Topology
same_as:
  wikidata: Q12510
  wikipedia: 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|0V8046R0P9]], is the torus $T^2$.

## Remarks

The torus is homeomorphic to the product $S^1 \times S^1$ of two [[circles|0WH1GVBJNH]], with the [[product topology|0Z57EC6YX3]], 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.