Nearnesses of finite order and uniqueness conditions for associated compactifications

Nearnesses of finite order are defined, and in particular a sub-class (called Ivanov n-nearnesses) which includes the class of Lodato proximities and leads to contiguities as a limiting case. A canonical compactification by maximal clans is constructed and characterised descriptively, thus unifying the compactification theories of I.O-proximities and of contiguities. Finally, it is shown how all maximal clans with respect to the finest contiguity compatible with a given Ivanov n-nearness can be constructed from prime closed filters, using not more than n such filters for each clan.