Metric Space is Open and Closed in Itself
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Theorem
Let $M = \struct {A, d}$ be a metric space.
Then $A$ is both open and closed in $M$.
Proof
From Metric Space is Open in Itself, $A$ is open in $M$.
From Metric Space is Closed in Itself, $A$ 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