Example

Let R denote the equivalence relation $x \equiv y \pmod{1}$ on the real line $\mathbf{R}$ (no. 4, Example) and let A denote the closed interval $[0, 1]$; A contains at least one point of each equivalence class mod R. The canonical mapping of $A/R_A$ onto the torus T is a homeomorphism; for if F is closed in A (and hence in R), in order to saturate F with respect to the relation R we have to take the union of the closed sets $F + n$ (for all $n \in \mathbf{Z}$), which evidently form a locally finite family, so that their union is closed (§ 1, no. 5, Proposition 4); the assertion follows from this. We remark that $A/R_A$ is obtained by identifying the points 0 and 1 in A. \*