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. 
 A: Hint:Let $C=(C_i)_{i\in I}$.Let $A=\cup_{i\in I} C_i$.
Now,(by contrdiction) suppose that $A$ is not connected.Then there is a function $g:A\to ${$0,1$} which is continuous and onto. Let $x_0\in \cap_{i\in I} C_i$. Then $x_0\in A$. Suppose that $g(x_0)=0$. Because $A$ is disconnected,there is a $x_1\in A:g(x_1)=1$.Also thre is a $i_1\in I:x_1\in C_{i_1}$.So $x_0,x_1\in C_{i_1}$.....your turn:)
A: 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 .
