# Definition:Closed Set/Normed Vector Space

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## Definition

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.

## Also see

## Sources

- 2017: Amol Sasane:
*A Friendly Approach to Functional Analysis*: $\S 1.3$: Normed and Banach spaces. Topology of normed spaces