Empty Set is Closed/Metric Space

Theorem
Let $M = \left({A, d}\right)$ be a metric space.

Then the empty set $\varnothing$ is closed in $M$.

Proof
From Metric Space is Open in Itself, $A$ is open in $M$.

But:
 * $\varnothing = \complement_A \left({A}\right)$

where $\complement_A$ denotes the set complement relative to $A$.

The result follows by definition of closed set.