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 $A=\{\{0\},[0,1],\{2\},[2,3]\}$. Then the union of its elements is $[0,1]\cup[2,3]$, which is not connected.