Remark 2

\* As we shall see in Chapter VI, § 2, no. 4, the topological space $\mathbf{T}$ is homeomorphic to the circle $x^2 + y^2 = 1$ in the real number plane $\mathbf{R}^2$; the product space $\mathbf{T}^2$ is homeomorphic to a torus of revolution in $\mathbf{R}^3$ (Chapter VII, § 1, Exercise 15). This is the origin of the name "one-dimensional torus" for $\mathbf{T}$ (in Chapter VII, § 1, we shall call $\mathbf{T}^n$ the n-dimensional torus) *.