Definition 1

The topological group $\mathbf{R}/a\mathbf{Z}$ ($a > 0$) is called the additive group of real numbers modulo $a$. The topological group $\mathbf{R}/\mathbf{Z}$ is denoted by $\mathbf{T}$. As a topological space, $\mathbf{T}$ is called the one-dimensional torus (by abuse of language, the topological group $\mathbf{T}$ is also called the one-dimensional torus).