The interior of a finite intersection is the intersection of the interiors

up:: Interior (Topology)

Let be a Topological Space.
Proof by induction:

1. Let . Note that, since All sets contain their interior, we have that , which implies that
2. Let . Then, as above,

Counterexample of interior of infinite intersections

Let (sets around )

• whose infinite intersection is
• , whose interior is

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References

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