Why are clopen sets a union of connected components? The wikipedia page on clopen sets says "Any clopen set is a union of (possibly infinitely many) connected components." 
I thought any topological space is the union of its connected components? Why is this singled out here for clopen sets?
Does it have something to do with it $x\in C$ a clopen subset $C$ of a space $X$, then $C$ actually contains the entire component of $x$ in $X$?

My attempt at a proof: suppose $S$ is clopen in $X$. If $X$ is connected we are done hence suppose $X=\bigcup_i C_i$ is the partition of $X$ by its connected components. We need to show that for each $i$ $C_i\cap S$ is either void or equal to $C_i$. But since $S$ is clopen, $S\cap C_i$ is both open and closed in $C_i$. Since $C_i$ is connected this implies either $S\cap C_i=\emptyset$ or $S\cap C_i=C_i$.
