\import{set.tex}

\section{Preclosure spaces}


\begin{struct}
    A preclosure space $X$ is a onesorted structure equipped with
    \begin{enumerate}
        \item $\cl$
    \end{enumerate}
    such that
    \begin{enumerate}
        \item For all $Y\subseteq X$ we have $\cl(Y)\subseteq X$.
        \item $\cl(\emptyset) = \emptyset$.
        \item For all $A$ we have $A\subseteq \cl(A)$.
        \item For all $A, B$ we have $\cl(A\union B) = \cl(A) \union \cl(B)$.
    \end{enumerate}
\end{struct}