# Closure of Topological Closure equals Closure

## Theorem

Let $T$ be a topological space.

Let $H \subseteq T$.

Then:

- $\left({H^-}\right)^- = H^-$

where $H^-$ denotes the closure of $H$.

## Proof

It follows directly from Set is Subset of its Topological Closure that:

- $H^- \subseteq \left({H^-}\right)^-$

$\Box$

Let $x \in \left({H^-}\right)^-$.

Then from Condition for Point being in Closure, any $U$ which is open in $T$ such that $x \in U$ contains some $y \in H^-$.

If we consider $U$ as an open set containing $y$, it follows that:

- $U \cap H \ne \varnothing$

Hence $x \in H^-$.

$\blacksquare$

## Sources

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $3.7$: Definitions: Proposition $3.7.15 \ \text{(c)}$