Subset of Standard Discrete Metric Space is Neighborhood of Each Point

Theorem
Let $M = \left({A, d}\right)$ be a metric space where $d$ is the standard discrete metric.

Let $S \subseteq A$.

Let $a \in S$.

Then $S$ is a neighborhood of $a$.

That is, every subset of $A$ is a neighborhood of each of its points.

Proof
Let $S \subseteq A$.

Let $a \in S$.

From Neighborhoods in Standard Discrete Metric Space, $\left\{{a}\right\}$ is a neighborhood of $a$.

As $a \in S$ it follows from Singleton of Element is Subset that $\left\{{a}\right\} \subseteq S$.

The result follows from Superset of Neighborhood in Metric Space is Neighborhood.