> [!quote] Idea > > Boundary is what lies between the interior and the exterior. Points that cannot be grouped into a set, but also cannot be separated from it. > [!definition] > > Let $(X, \topo)$ be a [[Topological Space|topological space]] and $A \subseteq X$. The **boundary** of $A$, > $ > \partial A = \ol{A} \setminus A^o = \ol{A} \cap \ol{A^c} > $ > is the difference between its [[Topological Closure|closure]] and its [[Interior|interior]]. > > *Proof*. Since $A^o \subset A$, $(A^o)^c$ is a [[Closed Set|closed set]] that contains $A^c$. Therefore $(A^o)^c \supset \ol{A^c}$. For any closed set $B \supset A^c$, $B \setminus A^o$ is another closed set that contains $A^c$. Therefore $\ol{A^c} \subset (A^o)^c$.