Point in Standard Discrete Metric Space is Isolated

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

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

Let $\alpha \in H$.

The $\alpha$ is an isolated point of $H$.

Proof
By definition of the standard discrete metric:


 * $\map d {x, y} = \begin {cases}

0 & : x = y \\ 1 & : x \ne y \end {cases}$

Let $\alpha \in H$.

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

Thus:

Hence the result by definition of isolated point.