# Union of connected sets with possible empty intersection

There is a result which states that if a collection $$A$$ of connected sets has a point $$P$$ belonging to every of those sets, then its union is connected

I was wondering if this remains true if the hypothesis is that for each of those sets $$X$$ there exists another one $$Y_X \in A$$ of then such that the intersection of those two is non-empty. With this condition can we conclude that the union is connected?

• No. Take $A=[0,1], B=[1,2],C=[3,4],D=[4,5]$. The union is not connected. – Mark May 18 at 16:41
• However, this may sometimes be useful: If the graph with the given connected sets as vertices and an edge between two sets iff they intersect is connected, then the union of our sets is connected ... – Hagen von Eitzen May 18 at 16:44

## 1 Answer

No. Take $$A=\{\{0\},[0,1],\{2\},[2,3]\}$$. Then the union of its elements is $$[0,1]\cup[2,3]$$, which is not connected.