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. 

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:)
