The boundary of the boundary is the boundary iff the boundary's interior is empty

up:: Boundary (Topology)

The boundary has no boundary if it doesn’t have any “meat”/“width” (i.e. no interior).

Let be a subset of a Topological Space .

Let be its Boundary. Then its boundary will be

Since The boundary is a closed set, this is equal to

Since the boundary has an empty interior by hypothesis, this yields

Suppose . Then

which can only hold if ─because else, we’d only have that The boundary of the boundary is a subset of the boundary , differing by at least one point.

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References

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