Category:Open Sets (Normed Vector Spaces)

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This category contains results about Open Sets in the context of Normed Vector Spaces.

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

Let $U \subseteq X$.

Then $U$ is an open set in $V$ if and only if:

$\forall x \in U: \exists \epsilon \in \R_{>0}: \map {B_\epsilon} x \subseteq U$

where $\map {B_\epsilon} x$ is the open $\epsilon$-ball of $x$.