Definition of closure from here
Boundary (Topology): The points in the closure of a set which are not in the interior of that set.
Definition of Exterior from here
Let $T$ be a topological space. Let $H \subseteq T$. The exterior of $H$ is the complement of the closure of $H$ in $T$. Alternatively, the exterior of $H$ is the interior of the complement of $H$ in $T$.