Subset of Standard Discrete Metric Space is Neighborhood of Each Point
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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 $1$