Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Serdica Mathematical Journal, Vol. 24, No 1, (1998), 99p-126p
In this paper the notion of SR-proximity is introduced and in
virtue of it some new proximity-type descriptions of the ordered sets of all
(up to equivalence) regular, resp. completely regular, resp. locally compact
extensions of a topological space are obtained. New proofs of the Smirnov
Compactification Theorem  and of the Harris Theorem on regular-closed
extensions [17, Thm. H] are given. It is shown that the notion of SR-proximity
is a generalization of the notions of RC-proximity  and Efremovicˇ proximity .
Moreover, there is a natural way for coming to both these notions starting
from the SR-proximities. A characterization (in the
spirit of M. Lodato [23, 24]) of the proximity relations induced by the regular
extensions is given. It is proved that the injectively ordered set of all
(up to equivalence) regular extensions of X in which X is 2-combinatorially
embedded has a largest element (κX, κ). A construction of κX is proposed.
A new class of regular spaces, called CE-regular spaces, is introduced; the
class of all OCE-regular spaces of J. Porter and C. Votaw  (and, hence,
the class of all regular-closed spaces) is its proper subclass. The CE-regular
extensions of the regular spaces are studied. It is shown that SR-proximities
can be interpreted as bases (or generators) of the subtopological regular
nearness spaces of H. Bentley and H. Herrlich .
∗ This work was partially supported by the National Foundation for Scientific Researches at the Bulgarian Ministry of Education and Science under contract no. MM-427/94.