Examples of closure operators from logic.
Take set of all propositions as a preorder, where \(p \leq q\) iff \(p\) implies \(q\).
Some modal operators are closure operators
E.g. \(j(p)\) means “Assuming Bob is in Chicago, p”
\(p \implies j(p)\) (the logic is monotonic, adding assumptions does not make something true into something false.
\(j(j(p)) = j(p)\) (the assumption is idempotent)