Exercise 1, § 5

a) Show that an equivalence relation $R$ on a topological space $X$ is open if and only if, for each subset $A$ of $X$, the saturation of the interior of $A$ is contained in the interior of the saturation of $A$. Give an example where $R$ is open and these two sets can be distinct.

b) Show that an equivalence relation $R$ on a topological space $X$ is closed if and only if, for each subset $A$ of $X$, the saturation of the closure of $A$ contains the closure of the saturation of $A$. Give an example where $R$ is closed and these two sets can be distinct.