Regular and Other Kinds of Extensions of Topological Spaces

Dimov, G.

Serdica Mathematical Journal (1998)

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

Abstract

top
∗ 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].

How to cite

top

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 -

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.