The boundary of the boundary is a subset of the boundary

up:: Boundary (Topology)

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

Let $\partial S$ be its Boundary. Then the boundary of the boundary of $S$ is

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

Since The boundary is a closed set, we have that

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

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

---

References

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