Example

*Consider the equivalence relation $x \equiv y \pmod{1}$ on the real line $\mathbf{R}$; the quotient space of $\mathbf{R}$ by this relation is called the one-dimensional torus and is denoted by $T$. The equivalence class of a point $x \in \mathbf{R}$ consists of all the points $x + n$, where $n$ runs through the set $\mathbf{Z}$ of rational integers. By Proposition 6 there is a one-to-one correspondence between the continuous functions on $T$ and the continuous functions on $\mathbf{R}$ which are periodic with period 1. We shall return to this important example in Chapter V, § 1.*