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.

Tag	`03NM7DZXXC`
URI	`m2://03NM7DZXXC`
Kind	example
Writing	W2 usable
MSC	57K20
Instance of	Topological space
Uses	Quotient topology, Product topology, Circle as a quotient of the real line
Same as	Wikidata Q12510 Wikipedia Torus

Tag

03NM7DZXXC

URI

m2://03NM7DZXXC

Kind

example

Writing

W2 usable

MSC

57K20

Instance of

Topological space

Uses

Quotient topology, Product topology, Circle as a quotient of the real line

Same as

Wikidata Q12510

Wikipedia Torus

Torus

Example

Remarks

Reading path

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Further reading

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