Empty Set is Closed/Metric Space
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Theorem
Let $M = \struct {A, d}$ be a metric space.
Then the empty set $\O$ is closed in $M$.
Proof
From Metric Space is Open in Itself, $A$ is open in $M$.
But:
- $\O = \relcomp A A$
where $\complement_A$ denotes the set complement relative to $A$.
The result follows by definition of closed set.
$\blacksquare$
Sources
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 6$: Open Sets and Closed Sets