The boundary of the boundary is a subset of the boundary

up:: Boundary (Topology)

Let be a subset of a Topological Space .

Let be its Boundary. Then the boundary of the boundary of is

Since The boundary is a closed set, we have that

For equality, we have that The boundary of the boundary is the boundary iff the boundary’s interior is empty.

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References

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