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 $X$?
Best Answer
Here is what I mean: If $C$ is clopen subset of a space $X$, then $C=\cup_{x\in C}C_x$, where $C_x$ is the connected component in $X$ containing $x$.
It is clear that $C\subset \cup_{x\in C}C_x$.
Let $x\in C$, and we want to show that $C_x\subset C$. Now we take $A=C_x\cap C$ and $B=C_x\cap C^c$, we have $A\cup B=C_x$ and $A\cap B=\emptyset $ and $A$, $B$ are closed subsets in $C_x$ hence opens subsets in $C_x$, since $C_x$ is connected then $A=\emptyset$ or $B=\emptyset$, but $x\in A$, so $B=\emptyset$, this show that $A=C_x$, $i.e$ $C_x\subset C$.