Proposition 6

Let $R$ be an equivalence relation on a topological space $X$; then the following three conditions are equivalent:

a) The relation $R$ is open.

b) The interior of each subset which is saturated with respect to $R$ is saturated with respect to $R$.

c) The closure of each subset which is saturated with respect to $R$ is saturated with respect to $R$.

By taking complements ($§ 1$, no. 6, formula (2)) we see that b) and c) are equivalent. Let us show that b) implies a): suppose condition b) is satisfied and let $U$ be an open subset of $X$, $V$ its saturation with respect to $R$; then $\dot{V} \supset U$, and since by hypothesis $\dot{V}$ is saturated, it follows that $\dot{V} = V$, and therefore the saturation of $U$ is open. Conversely, suppose condition a) is satisfied, and let $A$ be a saturated set; if $B$ is the saturation of $\dot{A}$, then $\dot{A} \subset B \subset A$, and since $B$ is open by hypothesis it follows that $B = \dot{A}$.