Added 5 August 2017

Professors who like to occasionally throw curveballs at students on Ph.D. quals might find a few in my answer below. While some partitions are trivial and most are easy to find, others are more challenging. For example, here is a moderately difficult one:

Problem. Give an example of a partition $\mathscr{P}=\{F_1,F_2,G_1,G_2,G_3\}$ of $\mathbb{R}$ under the usual topology such that $\varnothing\not\in\mathscr{P}$, each $F_i$ is closed, each $G_i$ is open, and $\overline{G_i}=G_i\cup F_1\cup F_2$ for $i=1,2,3.$

Solution. See space number 19 in the list of five-point spaces in my answer.

All of the evidence so far points to the truth of the conjecture. Short of finding a proof, it might be interesting to look for counterexamples in weakened spaces, for example in some other infinite connected space besides $\mathbb{R}.$ Or it might be interesting to replace $\mathbb{R}$ with finite spaces.

One more question. Many spaces in my answer seem to require the “middle thirds” structure of the Cantor construction in their associated real partitions. Many other spaces clearly do not. Assuming that some do in fact require it, what is it about them that makes this so?

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?

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?

Added 22 July 2017

Taking into account the path (described below) that led me to the conjecture, instead of restricting to just one unary operation $u$ that is arbitrarily defined on $\mathscr{P}(S)\setminus P$, it might be better to allow any given collection of operations (of whatever arity), some or all of which may be required to satisfy various axioms (such as for example, closure axioms), with the partition condition applying to each.

But the meat of this post is supposed to be the conjecture, so if the above framework is too general to be of any use (it may be), then let's just scrap it.

I arrived at the conjecture as follows.

Back in grad school in the mid-1980s, the Kuratowski closure-complement-intersection problem (a good discussion can be found here) attracted my attention. I noticed that one could easily write a program to generate families of sets of reals under the usual topology (from a single seed set) by partitioning and working solely within the partition—provided it satisfies the above condition for the closure operation on $\mathbb{R}$ (the condition holds trivially for the complement operation).

In 2009 I posted a Javascript app here that lets users vary the seed set in a ten-point quotient of the reals to see what family it generates under closure and complement. About a year ago, I became interested in the actual space “under the hood” of this app.

A natural question arose: since connected seven-point spaces exist that contain Kuratowski 14-sets, might it be possible to replace my ten-point quotient with a seven-point one? (The answer is yes.) Further exploration led me to the conjecture, which I am currently about halfway finished verifying for all 94 connected five-spaces (it holds for all smaller ones). This is where I stop...there are too many connected six-spaces!

As one would guess, Cantor sets play a prominent role in many of the partitions (roughly half).