Connected Subsets If C is a collection of connected subsets of M, all having a point in
common, prove that union of C is connected.
I know a set is connected if it is not disconnected. Also, from the above, I know the intersection of all subsets C is nonempty. I am not sure where to go from there. 

Let's take $U,V \in C$ now $U \cap V = \{ x\} $ now assuminig that $U \cup V $ is not connected which means that there are distincts $A,B$ such that $A \cup B =U \cup V$
so $A \subset U$ or  $A \subset V$ and $B \subset U$ or  $B \subset V$ it also cant be that both $A, B \subset U$  so without lost of generality  $A \subset U$ and  $B \subset V$ and this is acontadiction for A and B must be distinct.
For now this proves finite number of unions.
Now  X is connected if and only if for everey $a,b \in X$ there finite conected subspace which
$A_i \cap A_{i+1} \neq \emptyset$ and $a \in A_1 ,b \in A_n$
From these lema $X = \bigcup_{i\in I} A_i $ is conected .
