Definition
Consider a subset of a Topological Space . The boundary of is defined as
\partial A = \operatorname{bd}A &:= \bar{A} \setminus \operatorname{int}A \\ &= \bar{A} \cap \overline{(X \setminus A)} \\ &= X \setminus [\operatorname{int} A \cup \operatorname{int}(X\setminus A)] \end{aligned}$$ The boundary of the $A$ can be defined in any of the following equivalent ways: - $\partial A$ is the [[Closure]] of $A$ minus the [[Interior]] of $A$ in $X$. - $\partial A$ is the intersection of the [[Closure]] of $A$ with the [[Closure]] of its complement.