Suppose we have a partition $P$ of a set $S$ and a unary set operation $u:\mathscr{P}(S)\to\mathscr{P}(S)$ such that for each $A\in P$ the set $uA$ is saturated with respect to $P$ (a union of elements in $P$).

Can anyone suggest any references on this topic? So far the closest thing I have found is a 2013 paper by Christian Ronse entitled "Closures on partial partitions from closures on sets."

Here is the specific question I am curious about:

Conjecture. Let $X$ be a connected finite topological space. Let $u$ be the closure operation on $\mathbb{R}$ under the usual topology. Then there exists a partition $P$ of $\mathbb{R}$ satisfying the condition stated above, such that the quotient $\mathbb{R}/P$ is homeomorphic to $X$.

Even if this an intractable problem, it can be fun to search for counterexamples. Certain spaces are difficult to find a corresponding real partition (of the type above) for; some have had me thinking they were counterexamples for sure, then suddenly I find a partition that works.

Can anyone suggest any leads?