Point in Metric Space has Neighborhood

Theorem

Let $M = \struct {A, d}$ be a metric space.

Let $a \in A$ be a point in $M$.

Then there exists some neighborhood of $a$ in $M$.

Proof

Let $a \in A$.

Then $A$ is a neighborhood of $a$ in $M$.

$\blacksquare$

Sources

• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 4$: Open Balls and Neighborhoods: Theorem $4.8: \ N \, 1$