The boundary is a closed set

up:: Boundary (Topology)

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

Then its boundary is

$$\partial S=\overline{S}\setminus \mathring{S}=\overline{S}\cap {\mathring{S}}^C$$

Since the Complement of an open set is a closed set ─ and the Closure is a closed set, and the Interior is an open set─, we have an intersection of closed sets, which yields a closed set.

Thus, the boundary of a set is closed.