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.