# Empty Set is Open in Neighborhood Space

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## Theorem

Let $\struct {S, \NN}$ be a neighborhood space.

Then the empty set $\O$ is an open set of $\struct {S, \NN}$.

## Proof

Suppose $\O$ were not an open set of $\struct {S, \NN}$.

Then $\exists x \in \O$ such that $\O$ is not a neighborhood of $\O$.

By definition of empty set, such an $x$ does not exist.

Hence the result.

$\blacksquare$

## Sources

- 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $3$: Topological Spaces: $\S 3$: Neighborhoods and Neighborhood Spaces: Lemma $3.6$