Boundary

For a subset $𝑆$ of a topological space $𝑋$ , the boundary $𝜕 𝑆$ is the set of points that are neither the interior nor the exterior, topology i.e. points in $𝑋$ such that every open neighbourhood intersects both $𝑆$ and $𝑋 ∖ 𝑆$ . Equivalently, it is the closure minus the interior.

Using this definition, a set is open iff. it does not contain any of its boundary points.