Topological Closure is Closed

Theorem

Let $T$ be a topological space.

Let $H \subseteq T$.


Then the closure $\map \cl H$ of $H$ is closed in $T$.


Proof

From Closure of Topological Closure equals Closure:

$\map \cl {\map \cl H} = \map \cl H$

From Set is Closed iff Equals Topological Closure, it follows that $\map \cl H$ is closed.

$\blacksquare$


Also see


Sources

  • 1953: Walter Rudin: Principles of Mathematical Analysis ... (previous) ... (next): $2.27 \, \text a$
  • 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.7$: Definitions: Proposition $3.7.15 \ \text{(d)}$
  • 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Point Sets: $9.$
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