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.

  • $\begingroup$ 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

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?


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