Finite Subset of Metric Space is Closed

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Theorem

Let $M = \struct {A, d}$ be a metric space.

Let $S \subseteq A$ be finite.


Then $S$ is closed in $M$.


Proof

From Metric Space is Hausdorff, $M$ is Hausdorff.

From Finite Subspace of Hausdorff Space is Closed, $S$ is closed.

$\blacksquare$