\section{Closure operators}
\[X \subseteq \mathrm{cl}(X)\]
\[\mathrm{cl}( \mathrm{cl}(X) ) = \mathrm{cl}(X)\]
\[X \subseteq Y \implies \mathrm{cl}( X ) \subseteq \mathrm{cl}(Y)\]


Closure operators appear in various fields of mathematics; here are some prominent examples.

\begin{itemize}
	\item Closure operator (topology)
	\item Linear span operator (linear algebra)
	\item Closed linear span operator (functional analysis)
	\item Convex hull operator (convex analysis)
	\item "Derivable formulae" operator (mathematical logic)
\end{itemize}