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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 $T$.

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In general, if $T$ is a topological space, then any set $H$ in $T$ partitions $T$ into three disjoint sets: the interior of $H$, the boundary of $H$, and the exterior of $H$. So the exterior only equals the boundary when both are empty, that is when all of $T$ is in the interior of $H$, i.e. when $H$=$T$.

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    @hhh If $A$ is a subset, and $T$ is the whole space, $T = cl(A) \cup ext(A) = int(A) \cup bd(A) \cup ext(A)$. $cl(A) = int(A) \cup bd(A)$, where all unions are disjoint.2011-02-10