Set is Closed iff Equals Topological Closure/Proof 1

Theorem

Let $T$ be a topological space.

Let $H \subseteq T$.

Then $H$ is closed in $T$ if and only if:

$H = \map \cl H$

Proof

Let $H'$ denote the derived set of $H$.

By Closed Set iff Contains all its Limit Points, $H$ is closed in $T$ if and only if $H' \subseteq H$.

By Union with Superset is Superset, $H' \subseteq H$ if and only if $H = H \cup H'$.

The result follows from the definition of closure.

$\blacksquare$

Sources

• 1953: Walter Rudin: Principles of Mathematical Analysis ... (previous) ... (next): $2.27 b$