2
$\begingroup$

I am considering $(X,d)$, which is a set with a discrete metric and $D \subset X$. With these conditions given, I am supposed to show that $\partial D = \emptyset $. Further I am supposed to describe all open (closed) sets in $(X,d)$.

I kinda have the idea of why $\partial D = \emptyset$. I think it has to do with the fact that the metric space is endowed with a discrete metric. But i certainly don't know where to start. This is also making the second part a bit difficult.

I'd be really happy if someone could help me.

$\endgroup$
  • $\begingroup$ Suppose you had a point $x$ in the closure of $D$ but not in $D$ itself. What that would mean in terms of distances between $x$ and points of $D$? $\endgroup$ – Andrea Mori Sep 2 '14 at 15:21
0
$\begingroup$

I would suggest you describe the open and closed sets in $(X,d)$ first. As a starting point, what are the open balls in this metric space?

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.