Question:

Define the n-dimensional torus $T^{n}$ for n $\in \mathbb{Z}$. What is $T^{1}$ better known as?

Recalling the definition of a 2-dimensional torus, the n-dimensional torus $T^{n}$ is the quotient space obtained from $\mathbb{R}^{n}$ by the relation x~y IFF $x-y \in \mathbb{Z}^{n}$. A 2-dimensional torus, is formed by gluing $\left [ 0,1 \right ] \times \left [ 0,1 \right ]$. A 1-dimensional torus certainly requires only $\left [ 0,1 \right ]$. Is it a circle?

Any help is appreciated.

Thanks in advance.

  • 1-dimensional torus T^1=R/Z – Adam Sep 5 '17 at 15:52
up vote 1 down vote accepted

By common definition, we have: $$T^n = \underbrace{S^1 \times S^1 \times \dots \times S^1}_{n-\text{times}}$$ So, $$T^1 = S^1,$$ the one-sphere; you are correct.

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