# Regular and Other Kinds of Extensions of Topological Spaces

Serdica Mathematical Journal (1998)

- Volume: 24, Issue: 1, page 99-126
- ISSN: 1310-6600

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topDimov, 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 -

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