Let A and B be connected subspaces of a topological space (X,$\tau$). If A,B are not disjoint, prove that the subspace A $\cap$ B is connected.
Using the definition of connected space is that the only clopen set is the space itself, I tried proof by contradiction. Let S ($\neq A\cap B$) be clopen subset of A $\cap$ B, then there exist $U_1,U_2\in\tau$ s.t. $A\cap B\cap U_1=S,A\cap B\cap U_2=(A\cap B)\setminus S$. So $A\cap B$ is contained in $U_1\cap U_2\in\tau$. Not sure if these are useful, please give me some ideas.
If $f(s)=e^{2 \pi i s}$ is the usual parametrization of the circle, then $A=\{f(s) : 0 \leq s \leq 1/2\}$ and $B=\{f(s) : 1/2 \leq s \leq 1\}$ are connected, but $A \cap B = \{\pm 1\}$ is not connected.
I think what you really want to prove is the following: If $A,B \subseteq X$ are connected and not disjoint, then $A \cup B$ is connected. Here is a quick proof for this: A space is connected if and only if every continuous map to the discrete space $\{0,1\}$ is constant. A continuous map $A \cup B \to \{0,1\}$ induces continuous maps $A \to \{0,1\}$ and $B \to \{0,1\}$ which agree on $A \cap B$. Both are constant, and their values agree because $A \cap B$ is non-empty. Hence, the whole map is constant.