Empty Set is Open and Closed in Metric Space

Theorem

Let $M = \struct {A, d}$ be a metric space.

Then the empty set $\O$ is both open and closed in $M$.

Proof

From Empty Set is Open in Metric Space, $\O$ is open in $M$.

From Empty Set is Closed in Metric Space, $\O$ is closed in $M$.

$\blacksquare$

Sources

• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 6$: Open Sets and Closed Sets