Let \(X\) be a topological space and \(A\subseteq X\). A point is in \(\partial A\) if and only if every neighbourhood of it contains a point in \(A\) and in \(X\setminus A\).
Proof
Assume there is a neighbourhood that does not contain a point in \(A\) or in \(X\A\). Then it is fully contained in one of these and therefore \(x\in \Int A\) or \(x\in \Ext A\).
On the contrary \(x\) is neither in \(\Int A\) nor in \(\Ext A\).