Open Ball is Neighborhood of all Points Inside

Theorem

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

Let $a \in A$.

Let $\map {B_\epsilon} a$ be an open $\epsilon$-ball of $a$ in $M$.

Let $x \in \map {B_\epsilon} a$.

Then $\map {B_\epsilon} a$ is a neighborhoods of $x$ in $M$.

Proof

From Open Ball of Point Inside Open Ball:

$\exists \delta \in \R: \map {B_\delta} x \subseteq \map {B_\epsilon} a$

Thus by definition $\map {B_\delta} x$ is a neighborhoods of $x$ in $M$.

$\blacksquare$

Sources

• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 4$: Open Balls and Neighborhoods: Lemma $4.5$