Definition:Torus (Topology)/General

Definition

Definition 1

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

Definition 2

The $n$-dimensional torus (or $n$-torus) $\Bbb T^n$ is defined as the space whose points are those of the cross product of $n$ circles:

$\Bbb T^n = \underbrace{\Bbb S^1 \times \Bbb S^1 \times \ldots \times \Bbb S^1}_{n \text{ times}}$

and whose topology $\tau_{\Bbb T^n}$ is generated by the basis:

$\BB = \set {U_1 \times U_2 \times \cdots \times U_n : U_1, U_2, \ldots, U_n \in \tau_{\Bbb S^1}}$

where $\tau_{\Bbb S^1}$ is the topology of the $1$-sphere.

Also see

• Equivalence of Definitions of Torus of Arbitrary Dimension

• Results about the torus of arbitrary dimension can be found here.