The interior is the complement of the closure of the complement

up:: Interior (Topology)

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

Then, since All sets contain their interior, $Z^{\circ }\subseteq Z$.

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

$$(Z^c{)}^{-}\subseteq (Z^{\circ }{)}^c$$

By flipping again with complements, we arrive at

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

which shows an open set¹, which contains $Z$‘s interior. However, since $Z^{\circ }$ is the largest open set contained in $Z$, then we have that

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

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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. ↩