Clopen set

Remarks

Example: let $X = [0, 1] \cup [2, 3]$ with the topology from $\R$. The set $A = [0, 1]$ equals $X \cap (-1, 3/2)$, so it is open in $X$. Its complement $[2, 3]$ equals $X \cap (3/2, 4)$, so it is open as well, and $A$ is closed.

Example: in the rationals $\Q$ with the topology from $\R$, the set $\set{q \in \Q : q < \sqrt{2}}$ is clopen, since $\sqrt{2} \notin \Q$.

Non-example: in $\R$, the set $[0, 1)$ is not open, because no interval around $0$ lies inside it. It is not closed either.

In every space, $\emptyset$ and $X$ are clopen. A space $X$ is connected if and only if these are its only clopen subsets.