Let δ is a proximity.


I will call a set A connected iff for every partition {X,Y} of the set A holds X δ Y.


Question: Let A and B are sets with non-empty intersection. Let both A and B are connected. Prove or give a counter-example that A∪B is also connected.


(This question arouse as a special example of a more general theorem. I spend may be half of hour attempting to prove it and after these my efforts failed, I desire to share this question.)



Connectedness of a union regading a proximity
Consider $X\cap A$ and $Y\cap A$, starting from a partition $\lbrace X,Y\rbrace$ of $A\cup B$. If both intersections are nonempty we are done, as $(X\cap A)\delta(Y\cap A)$. Otherwise, $A\subseteq X$, say, but then $X\cap B$ and $Y\cap B$ are nonempty and we find $(X\cap B)\delta(Y\cap B)$.
In either case $X\delta Y$. 


