Let \(X\) be a topological space and \(A\subseteq X\). The boundary of \(A\) is defined by
\begin{equation*} \partial A = X\setminus (\Int A \cup \Ext A). \end{equation*}
Remarks
- \(\partial A\) is closed.
- A point is in \(\partial A\) if and only if every neighbourhood of it contains a point in \(A\) and in \(X\setminus A\).
- \(A\) is open if and only if it contains none of its boundary points.
- \(A\) is closed if and only if it contains all of its boundary points.
- \(\bar{A}=\Int A\cup \partial A\)