Metric Space is Closed in Itself
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Theorem
Let $M = \struct {A, d}$ be a metric space.
Then $A$ is closed in $M$.
Proof
From Empty Set is Open in Metric Space, $\O$ is open in $M$.
But:
- $A = \relcomp A \O$
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