Interior (Topology)

up:: Topological Space

Given a topological space $(X,\tau )$ and a set $Z\subset X$, we denote its interior as the union of all open sets contained inside it.

It’s essentially “inflating $Z$ from the inside-out” with open sets.

$$\mathring{Z}:=\bigcup _{\begin{array}{c}A\in \tau \\ A\subset Z\end{array}}A$$

Properties

• Interior preserves subset ordering
• All sets contain their interior
• The interior of an arbitrary union contains the union of the interiors
• The interior of a finite intersection is the intersection of the interiors

Interiors from Closures

Given a “closure operator”, one can infer its interior, since The interior is the complement of the closure of the complement.

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References

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