Example 1

Let $X$ be a topological space, $\Gamma$ a group of homeomorphisms of $X$ onto itself, and let $R$ be the equivalence relation

"there exists $\sigma \in \Gamma$ such that $y = \sigma(x)$"

between $x$ and $y$ (thus $R$ is the equivalence relation whose classes are the orbits of $\Gamma$ in $X$. The relation $R$ is open, for the saturation of a subset $A$ of $X$ with respect to $R$ is open so is each $\sigma(A)$ and hence so is their union.

\* On the real line $\mathbf{R}$ the equivalence relation $x \equiv y \pmod{1}$ is open, since it is derived as above from the group of translations

(see Chapter III, § 2, no. 4). \*