Open Ball is Open Set

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Theorem

Metric Space

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

Let $x \in A$.

Let $\epsilon \in \R_{>0}$.

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


Then $\map {B_\epsilon} x$ is an open set of $M$.


Normed Vector Space

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

Let $x \in X$.

Let $\epsilon \in \R_{>0}$.

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


Then $\map {B_\epsilon} x$ is an open set of $M$.