Boundary (Topology)

up:: 026 MOC Topology

Given a Topological Space $(X,\tau )$, the boundary of a set $Z$ \subset X is the set subtraction of its Closure by its Interior. This amounts to it consisting of the outermost points of $Z$.

$$\partial (Z)=\overline{Z}\setminus \mathring{Z}$$

Properties

• The boundary is a closed set
• The boundary of the set is the boundary of its complement
• The boundary of the boundary is a subset of the boundary
  • 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