Let \(X\) be a topological space and \(A\subseteq X\). The closure of \(A\) is defined by
\begin{equation*} \bar{A} = \cap \{A\subseteq B\mid B \text{ closed }\}. \end{equation*}
Remark
- \(\bar{A}\) is closed.
- \(A\subseteq \bar{A}\)
- \(A\subseteq B \implies \bar{A} \subseteq \bar{B}\)
- A point is in \(\bar{A}\) if and only every neighbourhood of it contains a point in \(A\).
- \(A\) is closed if and only if \(A=\bar{A}\).
- \(\bar{A}=\Int A\cup \partial A\)