Let \(X\) be a topological space and \(A\subseteq X\). The interior of \(A\) is defined by
\begin{equation*} \Int A = \cup \{B\subseteq A\mid B \text{ open }\}. \end{equation*}
Remarks
- \(\Int A\) is open.
- \(\Int A\subseteq A\)
- \(A\subseteq B \implies \Int A \subseteq \Int B\).
- A point is in \(\Int A\) if and only if it has a neighbourhood contained in \(A\).
- \(A\) open if and only if \(A=\Int A\).
- \(\bar{A}=\Int A\cup \partial A\)