Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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).

share|improve this question
    
Take the empty set. –  user127.0.0.1 Jan 15 at 0:39
    
ahh sorry should've said a nonempty subset –  user120246 Jan 15 at 0:44
    
I suppose the whole space is not allowed either? –  user127.0.0.1 Jan 15 at 0:49
1  
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
    
show 1 more comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.