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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.

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    What is a separation? – 2017-01-10
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    it is a collection of disjoint subsets of X whose union is X @MarianoSuárez-Álvarez – 2017-01-10
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    I'd say the usual.name for that is partition, and that in the context of general topology the word separation usually includes the condition that the sets be open. In any case, please add the intended meaning to the body of the question. – 2017-01-10
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    I mean partition sorry. – 2017-01-10
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    Yes yes I got confused with the English because in the terms of connectedness a separation must be open – 2017-01-10

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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.