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!
Best Answer
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 $E$.