The closure is the complement of the interior of the complement

up:: Closure (Topology)

Let $(X,\tau )$ be a Topological Space, and $Z\subseteq X$ a subset.

Then, since All sets are contained inside their closure, $Z\subseteq \overline{Z}$.

Since Set Complements flip subset ordering, we have that ${\overline{Z}}^c\subseteq Z^c$. Since $\overline{Z}$ is closed, ${\overline{Z}}^c$ is open, and equal to its own Interior. Thus,

$$({\overline{Z}}^c{)}^{\circ }={\overline{Z}}^c\subseteq (Z^c{)}^{\circ }$$

By flipping again with complements, we arrive at

$${\left[(Z^c{)}^{\circ }\right]}^c\subseteq \overline{Z}$$

which shows a closed set¹, which is contained inside $Z$‘s closure. However, since $\overline{Z}$ is the smallest closed set containing $Z$, then we have that

$${\left[(Z^c{)}^{\circ }\right]}^c=\overline{Z}$$

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References

• Notas para um Curso de Física-Matemática, João Carlos Alves Barata. Cap. 27

Footnotes

1. Since the complement of an open set/interior is a closed set. ↩