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!
$\endgroup$ 51 Answer
$\begingroup$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 $E$.
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