What is the smallest connected closed set? X is connected compact space , A is closed subset of X.
Is there the set B such that has the following property: 
1) B is  connected  closed set  contaning A
2) When C is proper subset of B and connected closed set, then C must not contain A
??
I think  there is definition  for such B 
Thank you for any help
 A: Such a set $B$ is not necessarily a smallest one.
Take for example the set $A$ as two disjoint closed disks in $[0,1]^2$. If we let $B$ be a set that contains $A$ and any line (that does not cross itself) connecting the disks in $A$, then this is an example of a minimal set as you describe. However, clearly we can take different lines to get different sets $B$.

If $X$ is Hausdorff, you can prove the existence of such a set. If $X$ is Hausdorff, we can use that the intersection of a chain of compact connected sets is connected
There always exists such a set $B$, since $X$ is connected. We can order closed connected subsets of $X$ that contain $A$ by inclusion, then every chain of such subsets has a lower bound, as the intersection of the chain will be closed, connected and contain $A$. The existence of $B$ is therefore guaranteed by Zorn's lemma.

A counterexample of a non-Hausdorff space is the line with two origins. If we let $A$ consist of these two origins, then taking intervals $[-\epsilon,\epsilon]$ for any $\epsilon>0$ contains $A$, but the intersection of all such intervals only contains $A$ and is not connected.
