Definition:Closed Set/Normed Vector Space

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

Let $F \subset X$.

Definition 1

$F$ is closed in $V$ if and only if its complement $X \setminus F$ is open in $V$.

Definition 2

$F$ is closed (in $V$) if and only if every limit point of $F$ is also a point of $F$.

That is: if and only if $F$ contains all its limit points.

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