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.



Union of proximally connected sets
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?)


