Definition:Torus (Topology)

Definition

A torus is a surface obtained by identifying both pairs of opposite sides, one with the other, of a square, while retaining the orientation:

Thus in the above diagram, $AB$ is identified with $DC$ and $CB$ with $DA$.

General Definition

The $n$-dimensional torus (or $n$-torus) $\Bbb T^n$ is defined as the $n$-fold product space of the $1$-sphere.

That is:

$\ds \Bbb T^n = \prod_{i \mathop \in \N_{< n}} \Bbb S^1$

where:

$\Bbb S^1$ denotes the $1$-sphere

$\N_{< n}$ denotes an initial segment of natural numbers

$\ds \prod_{i \mathop \in I}$ denotes the product space

Also see

• Results about the torus can be found here.

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.8$: Quotient spaces