Compactifications, \(A\)-compactifications and proximities (Q1913370)

A functor \(S\) from Alexandroff spaces \(X\) into distributive lattices is studied and used to describe proximities on \(X\) and an isomorphism between all compactifications of \(X\) and some sublattices of \(S(X)\). At the end, two concrete categories are introduced, one of them isomorphic and the other dual to \textbf{Prox} (objects of the categories are pairs composed of Alexandroff spaces and lattices, or by certain frames and lattices, resp.).