The theory of the
compactifications of a completely regular space has been elucidated in recent years by
the theory of proximities, introduced by Efremovič and developed especially by
Smirnov. The two fundamental results are that a space has a compactification if and
only if it has the topology of some proximity, and that there is a one-to-one
correspondence from the collection of compactifications of a space onto the collection
of proximities that give the topology of the space. We shall generalize these results
to a larger class of spaces, which are related to the regular-closed spaces
in the same manner as completely regular spaces are related to compact
spaces.
