Relation between clopen subset and connected subset of a topological space Let $X$ be a topological space. Suppose $A$ is clopen in $X$ and $C$ is a connected subset of $X$. If $A \cap C \neq \emptyset$, show that $C \subseteq A$.
  I tried proceeding by contradiction: suppose $C\nsubseteq A$. Then this means that there is a point $x \in C$ such that $x\in A$ and $x \in X\setminus A$.
 I am not sure what to do next.

Since $A$ is clopen, $A$ and $X \setminus A$ form a decomposition of $X$ into two open subsets.
Each of those two sets has nonempty intersection with $C$. 
So $C$ is not connected.
