Interesting orderings open sets of a topology Let X be a set of points an $\mathcal{Q}$ be a partition on X. The intuition I want to model is that X is a set of worlds considered possible by an agent and $\mathcal{Q}$ is a question whose partition cells are answers to the question.
Now I have several questions:


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*Let $\tau_Q$ be the partition topology on X generated by the basis $\lbrace A\subseteq X| A \in \mathcal{Q}\rbrace$. Is there an interesting way, in the sense of mathematically interesting and also relating to the intuition I want to model, how to order the open sets? It seems just taking subsets is not great because all the elements of the basis are going to be disjoint, but we want to order answers to a question somehow in the end. So if the "answers" are not comparable because they're all disjoint, that seems bad.
Another idea was to have first an ordering $\leq$ on the points ("worlds") and then say that, if $s,t\in X$, then $s \leq t$ iff. $A_s \sqsubseteq A_t$, where $A_s$ and $A_t$ are some kind of special neighbourhoods of $s$ and $t$ respectively.

*What other interesting topologies are out there if one has a set and a partition on the set?

*In general, what are interesting orderings of open sets in topological spaces?

 A: This answer concerns only question 2, which I originally thought should be easy because any such topology has to be invariant under all permutations of $X$ that leave the partition $\mathcal Q$ invariant. It turns out to be easy, but there are lots of cases, so let me give you a fairly typical example of what such a topology can look like. For brevity, I'll write "block" for "element of $\mathcal Q$", i.e., for an equivalence class of the associated equivalence relation.
The topology I have in mind has a basis consisting of the following sets.
(1) All the singletons $\{a\}$ such that the block that contain $a$ has cardinality an odd prime integer.
(2) All the blocks whose cardinalities are either finite and even or $\aleph_{12}$.
(3) All the sets $A$ that are subsets of blocks $B$ of cardinality $\aleph_7$ such that $B-A$ has cardinality $\aleph_3$.
(4) One more set, namely the union of all blocks not mentioned above, i.e., the union of all blocks of finite, odd, composite cardinalities and all infinite blocks of cardinalities other than $\aleph_7$ and $\aleph_12$.
Other topologies definable from a partition will be similar in that they will be generated by a basis with a description similar to some of these four clauses but with all the numbers and sets of numbers possibly altered. (Technically, to make this really correct, "cardinality $\aleph_3$" in clause (3) should be replaced with "cardinality $<\aleph_4$", so that it can be altered to, for example, "cardinality $<\aleph_\omega$".)
