# Empty Set is Open in Normed Vector Space

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## Theorem

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

Then the empty set $\O$ is an open set of $M$.

## Proof

By definition, an open set $S \subseteq X$ is one where every point inside it is an element of an open ball contained entirely within that set.

That is, there are no points in $S$ which have an open ball some of whose elements are not in $S$.

As there are no elements in $\O$, the result follows vacuously.

$\blacksquare$

## Sources

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