For [[knowledge-mathematics-analysis-functional-analysis-metric-spaces|metric space]] \((X,d)\) a boundary point for subset \(A\subseteq X\) is a point \(x\in X\) such that for all \(\epsilon >0\): 1. \(B_\epsilon(x)\cap A\neq\emptyset\) 2. \(B_\epsilon(x)\cap A^{c\neq}\emptyset\) The set of all boundary points is commonly denoted by \[ \partial A:=\{x\in X:x\text{ is a boundary point for }A\} \]
Note the following obvious results follow from the definition: 1. If \(A\subset X\) is an open set then \(A\cap\partial A=\emptyset\). 2. If \(A\subseteq X\) is a closed set then \(A\cap\partial A=A\).