Interior (Topology)

up:: Topological Space

Given a topological space and a set , we denote its interior as the union of all open sets contained inside it.

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

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