The \(n\)-torus \(\mathbb{T}^n\) is a topological group .
Proof
The unit circle
\(\mathbb{S}^1\subseteq \mathbb{C}\) is together with multiplication a topological group (see (0x67c2a266)
).
Since direct products of topological groups are again topological groups
, \(\mathbb{T}^n=\mathbb{S}^1\times \cdots \times \mathbb{S}^1\) is together with the direct product multiplication
a topological group.