Equivalence of neighborhood systems and preclosures (proof needed) I need a proof that neighborhood systems for a pretopology on a set $U$ bijectively correspond to preclosure operators on $U$ by the formulas:
$\begin{array}{lll}
  & \operatorname{cl} \left( A \right) = \left\{ x \in U \hspace{1em} | \hspace{1em}
  \forall X\in\Delta \left( x \right): X\cap A \ne \varnothing \right\} ; & \\
  & \Delta \left( x \right) = \left\{ A \in \mathscr{P} U \hspace{1em} |
  \hspace{1em} x \notin \operatorname{cl} \left( U \setminus A \right) \right\}  & 
\end{array}$
where $\operatorname{cl}$ is the closure operator and $\Delta(x)$ is the neighborhood filter for a point $x$.

It is now proved in my draft book.
