0
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ No. Take $A=[0,1], B=[1,2],C=[3,4],D=[4,5]$. The union is not connected. $\endgroup$ – Mark May 18 at 16:41
  • 1
    $\begingroup$ 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 ... $\endgroup$ – Hagen von Eitzen May 18 at 16:44
0
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.