two layer separation is homeomorphic to inverse of second layer

Question

Let $X$ be topological space, $S$ partition of X and $S'$ a partition of $X/S$.

Let $p:X\to X/S$ be the quotient map.

Show that there is a "natural" homeomorphism $(X/S)/S'\to X/T$, when $T$ is a partition of $X$ whose elements are of the form $p^{-1}(A)$ for $A\in S'$.

Edit:

It is easy to see that $T$'s elements are actually disjoint, and with some work I have proved that the union of all it's elements is X. Then we have that T is a partition of X.

Answer 1

I don't have time to post a full response right now, but this topological fact may be of interest to you:

Lemma: Given a map $f:X\to Y$, there is an induced map $\tilde f:(X/\mathord{\sim})\to Y$ where $\sim$ is the partition of $X$ given by $x\sim y$ if $f(x)=f(y)$. In particular, if $f$ is a surjection then $\tilde f$ is a continuous bijection, and if $f$ is a quotient map then $\tilde f$ is a homeomorphism.

I will try to update with a real answer later if nobody else has by then.