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 $X$?

Take the space $\mathbb{R}$. Then the subset $\mathbb{R}_{>0}$ of positive elements is a subset of $\mathbb{R}$, but is not a union of connected components of $\mathbb{R}$ (the only connected component of $\mathbb{R}$ is $\mathbb{R}$ itself). But then that's okay, because $\mathbb{R}_{>0}$ isn't clopen in $\mathbb{R}$ (it is only open).
What you seem to be doing here (I think) is to pass down to a subset of a topological space but then change the notion of connected component (perhaps to the connected components of the subspace with the subspace topology). What the above is saying is that if $U \subset X$ is a clopen subset of the space $X$, then it is a union of connected components of $\bf{X}$ (not of $U$).
