Open Ball of Point Inside Open Ball

Theorem

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

Let $\map {B_\epsilon} x$ be an open $\epsilon$-ball in $M = \struct {A, d}$.

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

Then:

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

That is, for every point in an open $\epsilon$-ball in a metric space, there exists an open $\delta$-ball of that point entirely contained within that open $\epsilon$-ball.

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

Let $\map {B_\epsilon} x$ be an open $\epsilon$-ball in $M = \struct {A, d}$.

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

Then:

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

That is, for every point in an open $\epsilon$-ball in a pseudometric space, there exists an open $\delta$-ball of that point entirely contained within that open $\epsilon$-ball.

Let $M = \struct {X, \norm {\, \cdot \,} }$ be a normed vector space.

Let $\map {B_\epsilon} x$ be an open $\epsilon$-ball in $M = \struct {X, \norm {\, \cdot \,} }$.

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

Then:

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

That is, for every point in an open $\epsilon$-ball in a normed vector space, there exists an open $\delta$-ball of that point entirely contained within that open $\epsilon$-ball.