Open Ball of Point Inside Open Ball

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Theorem

Metric Space

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.


Pseudometric Space

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.


Normed Vector Space

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.