Partition Topology I am trying to prove the following equivalence:
"Let $X$ be a set and $R$ be a partition of $X$, this is:
i) $(\forall A,B \in R, A \neq B) \colon A \cap B = \emptyset$
ii) $ \bigcup_{A \in R} A = X$
We say that a topology $\tau$ on $X$ comes from is a partition topology iff $\tau = \tau(R)$ for some partition $R$ of $X$.
Then a topology $\tau$ is a partition topology iff every open set in $\tau$ is also a closed set."
I am trying to proove $\Leftarrow$. I have tried using Zorn to proove the existance of a kind of maximal refinement of $\tau$ so as to find the partition that could generate $\tau$ but I am getting nowhere. I would truly appreciate any help posible...

Alternative hint: $R$ consists of the closures of the one-point sets.
