Empty Set is Closed/Metric Space

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
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