Set in Standard Discrete Metric Space has no Limit Points

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Let $M = \struct {S, d}$ be the standard discrete metric space on a set $A$.

Let $H \subseteq S$ be a subset of $S$.

Then $H$ has no limit points.


By definition of the standard discrete metric:

$\map d {x, y} = \begin {cases} 0 & : x = y \\ 1 & : x \ne y \end {cases}$

Let $\alpha \in S$.

By definition, $\alpha$ is a limit point of $H$ if and only if every deleted $\epsilon$-neighborhood $\map {B_\epsilon} \alpha \setminus \set \alpha$ of $\alpha$ contains a point in $A$:

$\forall \epsilon \in \R_{>0}: \paren {\map {B_\epsilon} \alpha \setminus \set \alpha} \cap H \ne \O$

Let $\map {B_1} \alpha$ be the open $1$-ball of $\alpha$ in $M$.


\(\ds \map {B_1} \alpha\) \(=\) \(\ds \set {y \in S: \map d {\alpha, y} < 1}\)
\(\ds \) \(=\) \(\ds \set \alpha\)
\(\ds \leadsto \ \ \) \(\ds \map {B_1} \alpha \setminus \set \alpha\) \(=\) \(\ds \O\)
\(\ds \leadsto \ \ \) \(\ds \paren {\map {B_1} \alpha \setminus \set \alpha} \cap H\) \(=\) \(\ds \O\) Intersection with Empty Set

So there exists a deleted $\epsilon$-neighborhood $\map {B_\epsilon} \alpha \setminus \set \alpha$ of $\alpha$ containing no points of $H$.

Hence $\alpha$ is not a limit point of $H$

As $\alpha$ is arbitrary, the result follows.