Metric Space is Closed in Itself

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Theorem

Let $M = \left({A, d}\right)$ be a metric space.

Then $A$ is closed in $M$.


Proof

From Empty Set is Open in Metric Space, $\varnothing$ is open in $M$.

But:

$A = \complement_A \left({\varnothing}\right)$

where $\complement_A$ denotes the set complement relative to $A$.

The result follows by definition of closed set.

$\blacksquare$


Sources