Subset of Standard Discrete Metric Space is Neighborhood of Each Point

Theorem

Let $M = \struct {A, d}$ 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, $\set a$ is a neighborhood of $a$.

As $a \in S$ it follows from Singleton of Element is Subset that $\set a \subseteq S$.

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

$\blacksquare$

Sources

• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 4$: Open Balls and Neighborhoods: Exercise $1$