Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $x$.

share|improve this question
add comment

1 Answer

up vote 0 down vote accepted

It is now proved in my draft book.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.