Compactifications, A-compactifications and proximities.

A functor S from the category Reg sigma Frm of regular sigma-frames to the category DLat of distributive lattices is defined. The notion of AP-sublattice of S(alpha), for alpha is an element of \Reg sigma Frm\, is introduced and it is shown that, for every Alexandroff space (X, alpha), the ordered set of all (up to equivalence) A-compactifications of (X, alpha) is isomorphic to the set of all AP-sublattices of S(alpha) ordered by inclusion. This gives, in particular, a description of the ordered set of z-compactifications of a T-3 1/2-space. Further for any Tychonoff space X the notion of a P-sublattice of S(CozX) is defined and it is proved that the ordered set of all (up to equivalence) T-2-compactifications of X is isomorphic to the set of all P-sublattices of S(CozX) ordered by inclusion. We construct proximities by means of AP- and P-sublattices. Moreover, using these notions, we introduce two concrete categories PHA and PASF which are respectively isomorphic and dual to the category Prox of proximity spaces.