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 $S\subseteq X$ be a subset of a Topological Space $(X,\tau )$.

Let $\partial S$ be its Boundary. Then its boundary will be

$$\partial \partial S=\overline{\partial S}\setminus \mathring{\partial S}$$

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

$$\partial \partial S=\partial S\setminus \mathring{\partial S}$$

$\mathring{\partial S}=\varnothing ⟹\partial \partial S=\partial S$

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

$$\partial \partial S=\partial S$$

$\partial \partial S=\partial S⟹\mathring{\partial S}=\varnothing$

Suppose $\partial \partial S=\partial S$. Then

$$\partial \partial S=\partial S\setminus \mathring{\partial S}=\partial S$$

which can only hold if $\mathring{\partial S}=\varnothing$ ─because else, we’d only have that The boundary of the boundary is a subset of the boundary $\partial \partial S\subset \partial S$, differing by at least one point.

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References

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