The boundary of the set is the boundary of its complement

up:: Boundary (Topology)

Let be a set in a Topological Space . Denote the Closure by and the Interior as .

Then its Boundary is

where is the set’s Complement.

Using that The closure is the complement of the interior of the complement and flipping the intersection, we have

Using that The interior is the complement of the closure of the complement yields

Thus, .