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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 defined as:

$U \in \tau_{\Bbb T^n} \iff \exists U_1, U_2, \ldots, U_n \in \tau_{\Bbb S^1} : U = U_1 \times U_2 \times \ldots \times U_n$

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