Let (δ;U) is a proximity space.
I will call a set A connected iff for every partition {X,Y} of the set A holds X δ Y.
Is the following true? (I need a proof or a counter-example.)
Conjecture If S is a collection of connected sets and ∩S≠∅ then ∪S is connected.
Note that instead of proximity we may consider the more general case of δ being limited only by the axioms (a subset of axioms of proximity space):
¬(∅δB), ¬(Aδ∅), X∪YδB⇔XδB∨YδB, AδX∪Y⇔AδX∨AδY.
Let $\{X,Y\}$ be a partition of the union of the family $S$, and let p be a point in the intersection of $S$ (which you've assumed is nonempty). Without loss of generality, p is in $X$. But, since a partition can't contain the empty set, $Y$ contains some point q from some set $A$ in the family $S$. Then $\{A\cap X, A\cap Y\}$ is a partition of $A$. By assumption, $A$ is connected, so $(A\cap X)\delta(A\cap Y)$. Therefore $X\delta Y$ as required. (All that was used about $\delta$ is that it's monotone in both arguments, which follows from your assumptions about unions.)
(Could this be homework? Do proximity spaces show up in classes?)
I don't know about the more general notion you've introduced, but this is definitely true for proximity spaces. A space is proximally connected iff every proximally continuous function from it to a discrete space is constant. So a family of proximally connected sets that all have a point in common, will have a proximally connected union, as any proximally continuous function from that union to a discrete space would have to have all values agreeing with the value on the common point.
See also this earlier question: Partition into connected sets by proximity
