Definition:Torus (Topology)

The $n$-dimensional torus $$\mathbb{T}^n$$ is defined as the space whose points are those of the cross product of $$n$$ circles:

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

and whose topology $$\vartheta_{\mathbb{T}^n}$$ is defined as

$$U \in \vartheta_{\mathbb{T}^n} \iff \exists U_1, U_2, \ldots, U_n \in \vartheta_{\mathbb{S}^1} | U = U_1 \times U_2 \times \ldots \times U_n$$

where $$\vartheta_{\mathbb{S}^1}$$ is the topology of the circle.