Define a subset of a metric space that is both open and closed. 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).

This question has been answered in comments:

Any set with the discrete topology is metrizable, and any subset of a discrete topological space is both open and closed, so...there you go! –  Nick D. Jan 15 at 0:50
Proving the every subset of $M$ is clopen. –  Poppy Jan 15 at 2:03

Note that:

$\mathbb{R}^n$ is connected, which means it cannot be written as the disjoint union of non-empty open sets. Equivalently, if $S\subseteq\mathbb{R}^n$ is open and closed, then either $S=\emptyset$ or $S=\mathbb{R}^n$. You have to find a topology where the underlying space is disconnected. –  T.A.E. Jan 15 at 5:48

