Regular and Other Kinds of Extensions of Topological Spaces

Dimov, G.. "Regular and Other Kinds of Extensions of Topological Spaces." Serdica Mathematical Journal 24.1 (1998): 99-126. <http://eudml.org/doc/11582>.

@article{Dimov1998,
abstract = {∗ 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.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 [31] 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 [17] and Efremovicˇ proximity [15]. 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 [29] (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 [4].},
author = {Dimov, G.},
journal = {Serdica Mathematical Journal},
keywords = {Regular; Regular Closed; Compact; Locally Compact; Completely Regular; CE-Regular; Extensions; SR– (R–, RC–, EF–) Proximities; Nearness Spaces; OCE– (CE–) Regular Spaces; regular-closed compact extensions; locally compact extensions; completely regular extensions; CE-regular extensions; SR-proximities; OCE-regular spaces; R-proximities; RC-proximities; EF-proximities; CE-regular spaces},
language = {eng},
number = {1},
pages = {99-126},
publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},
title = {Regular and Other Kinds of Extensions of Topological Spaces},
url = {http://eudml.org/doc/11582},
volume = {24},
year = {1998},
}

TY - JOUR
AU - Dimov, G.
TI - Regular and Other Kinds of Extensions of Topological Spaces
JO - Serdica Mathematical Journal
PY - 1998
PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences
VL - 24
IS - 1
SP - 99
EP - 126
AB - ∗ 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.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 [31] 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 [17] and Efremovicˇ proximity [15]. 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 [29] (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 [4].
LA - eng
KW - Regular; Regular Closed; Compact; Locally Compact; Completely Regular; CE-Regular; Extensions; SR– (R–, RC–, EF–) Proximities; Nearness Spaces; OCE– (CE–) Regular Spaces; regular-closed compact extensions; locally compact extensions; completely regular extensions; CE-regular extensions; SR-proximities; OCE-regular spaces; R-proximities; RC-proximities; EF-proximities; CE-regular spaces
UR - http://eudml.org/doc/11582
ER -