Open Ball is Open Set/Normed Vector Space

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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 $M$.


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

From Open Ball of Point Inside Open Ball in Normed Vector Space, there exists $\delta \in \R_{>0}$ such that $\map {B_\delta} y \subseteq \map {B_\epsilon} x$

The result follows from the definition of open set.