Metric Space is Open and Closed in Itself

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
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