Define a nonempty subset of a metric space that is both open and closed.
The real line with the Euclidean metric $d(x,y)=|x-y|$ is open and closed. If you take two real lines, not connected together, and invent a metric that works for any pair of points (it has to be able to give a distance if one point is on one line and one is on the other, as well as a distance between two points on the same line), then you have a nice disconnected metric space. And one of the lines is a closed open subset.
(Provided you can make sure there's a minimum distance between pairs of points on different lines)
I'm having some trouble with metric spaces and can't think of a subset that would be both open and closed (except for empty subset).