Assume we have a topological space $X $ and we have an equivalence relations $R,S $. We have the quotient spaces $(X/R)/S,(X/S)/R,X/(R\cap S) $. Are these spaces homeomorphic?
Note that this question is both set theoretical and topological. That is, is it true "set-wise" and if so, whether the identity is a homeomorphim.
I'm not sure if this question is trivial or does it require a proof using the universal property or so.