# 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).

• Take the empty set. Jan 15 '14 at 0:39
• ahh sorry should've said a nonempty subset Jan 15 '14 at 0:44
• I suppose the whole space is not allowed either? Jan 15 '14 at 0:49
• 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! Jan 15 '14 at 0:50
• math.stackexchange.com/questions/496235/… Jan 15 '14 at 2:03

Proving the every subset of $M$ is clopen. – Poppy Jan 15 at 2:03
$$\mathbb{R}^n$$ 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