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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 $x$.

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up vote 0 down vote accepted

It is now proved in my draft book.

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