Prove that a quotient topological space is Hausdorff if and only if any two distinct equivalence classes are contained in two disjoint open saturated sets.
This is a more detailed one that I've tried.
Let X be a space and ~ an equivalence relation on X. Let X/~ denote the set of all equivalence classes determined by ~. Let q be a quotient map of ~ such that . Now, the space induced by a quotient map q is a quotient space of X.
Suppose a quotient topological space is Hausdorff.
Let be distinct points in and U, V be disjoint open sets in containing and , respectively. By definition of a saturated set and Hausdorff property of , and are disjoint open saturated sets containing and in X, respectively. The and are simply equivalence classes in X determined by ~.
(If X/~ is Hausdorff, then X is Hausdorff. But the converse is not necessarily true).
Conversely, suppose any two distinct equivalence classes and in X determined by ~ are contained in two disjoint open saturated sets V and W in X.
Now, for any pair of distinct points and , we have two distinct points q(x) and q(y) in contained in two distinct open sets q(V) and q(W), respectively (The q maps saturated open sets of X to open sets of X/~).
Thus, is Hausdorff.