Exercise 5, § 5

Let $X$ be a topological space and let $R, S$ be two equivalence relations on $X$ such that $R$ implies $S$.

a) Show that if $S$ is open (resp. closed) then $S/R$ is an open (resp. closed) equivalence relation on $X/R$; show also that the converse is not necessarily true. Show that $S$ can be open (resp. closed) but not $R$.

b) Suppose that $R$ is open (resp. closed). Show that $S/R$ is open (resp. closed) if and only if $S$ is open (resp. closed).