The closure of a finite union is the union of the closures

up:: Closure (Topology)

Let be a Topological Space.
Proof by induction:

1. Let . Note that, since All sets are contained inside their closure, we have that , which implies that
2. Let . Then, as above,

Counterexample of closure of infinite unions

Let .

• whose infinite union is
• , whose closure is

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References

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