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I have been asked to provide examples (or proofs that none exist) regarding some points and subsets of discrete metric spaces. I believe I can set the interval/segment that the discrete metric space covers, I just have to provide a valid example that fits/shows the definition of these (among other) definitions:

1 - A neighborhood of a point p is a set Nr(P) consisting of all points q such that d(p, q) < r. The number r is called the radius of Nr(p).

2 - A point p is a limit point of the set E if every neighborhood of p contains a point q $\neq$ p such that q $\in$ E.

3 - E is closed if every limit point of E is a point of E.

Even if you cannot provide examples for all of the points and subsets, I would very much appreciate help with any of them. I can try to piece together the others. Also, just to clarify, there are several other definitions I have to provide examples for, these are just a few to help me better understand the concept.. Thanks for looking into my problem!

Andrés E. Caicedo
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user3495690
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    Hint: fix a point $p$ and consider the open and closed balls of radius $\frac12$. What does this tell you about the set ${p}$? – Math1000 Oct 01 '16 at 15:21
  • @Math1000 Is this in reference to number 2? From what I can gather, in a discrete metric space, there are no limit points. Which (of course) implies for number 3 that all of the limit points are contained in E (since there are none). Does that make sense? If so, would you mind doing me a huge favor and helping me put it into words that make sense for a math course? I think that's most of the problem I'm having.. – user3495690 Oct 01 '16 at 15:46
  • We have for each $p$: $$\left{q : d(p,q)<\frac12\right} = {p}.$$ The union of open sets is open, so therefore what can we conclude about the open sets in space? – Math1000 Oct 01 '16 at 15:48
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    That each point in the space is an isolated point? – user3495690 Oct 01 '16 at 16:05
  • @Math1000 You'll have to forgive me, I'm new to these concepts and trying to get a grasp on them. So I'm really not sure what to conclude from these hints.. – user3495690 Oct 01 '16 at 16:27

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If $E$ is a nonempty set, the map $$d(p,q)=\begin{cases} 1,&p\ne q\\0,& p=q\end{cases} $$ induces the discrete topology on $E$ - that is, every set of $E$ is open. For if $p\in Q$ then $$\left\{q\in E:d(p,q)<\frac12\right\}=\{p\} $$ is open, and thus for any $S\subset E$ we have $$S=\bigcup_{p\in S}\{p\}, $$ so that $S$ is open.

We also conclude that every point of $E$ is an isolated point, since e.g. the neighborhood of $p$ with radius $\frac12$ does not contain a point $q\ne p$. This means that every subset of $E$ is closed, since there are no limit points in $E$.

Math1000
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    Thank you! That makes a lot of sense.. I think I can get the rest of them with those examples. I was mostly having trouble putting things into words, and I wanted to be sure I understood metric sets properly before doing all of the questions. Which I believe I do, now! :) – user3495690 Oct 01 '16 at 16:39
  • I've found another problem I'm stuck on, if you don't mind giving me a hand... I'm trying to find an example of an interior point of a discrete metric space, but I don't fully understand what that entails. Do you mind helping me out? – user3495690 Oct 01 '16 at 17:09