Completely regular spaces

H. L. Bentley; Eva Lowen-Colebunders

Commentationes Mathematicae Universitatis Carolinae (1991)

  • Volume: 32, Issue: 1, page 129-153
  • ISSN: 0010-2628

Abstract

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We conduct an investigation of the relationships which exist between various generalizations of complete regularity in the setting of merotopic spaces, with particular attention to filter spaces such as Cauchy spaces and convergence spaces. Our primary contribution consists in the presentation of several counterexamples establishing the divergence of various such generalizations of complete regularity. We give examples of: (1) a contigual zero space which is not weakly regular and is not a Cauchy space; (2) a separated filter space which is a z -regular space but not a nearness space; (3) a separated, Cauchy, zero space which is z -regular but not regular; (4) a separated, Cauchy, zero space which is μ -regular but not regular and not z -regular; (5) a separated, Cauchy, zero space which is not weakly regular; (6) a topological space which is regular and μ -regular but not z -regular; (7) a filter, zero space which is regular and z -regular but not completely regular; and, (8) a regular Hausdorff topological space which is z -regular but not completely regular.

How to cite

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Bentley, H. L., and Lowen-Colebunders, Eva. "Completely regular spaces." Commentationes Mathematicae Universitatis Carolinae 32.1 (1991): 129-153. <http://eudml.org/doc/247257>.

@article{Bentley1991,
abstract = {We conduct an investigation of the relationships which exist between various generalizations of complete regularity in the setting of merotopic spaces, with particular attention to filter spaces such as Cauchy spaces and convergence spaces. Our primary contribution consists in the presentation of several counterexamples establishing the divergence of various such generalizations of complete regularity. We give examples of: (1) a contigual zero space which is not weakly regular and is not a Cauchy space; (2) a separated filter space which is a $z$-regular space but not a nearness space; (3) a separated, Cauchy, zero space which is $z$-regular but not regular; (4) a separated, Cauchy, zero space which is $\mu $-regular but not regular and not $z$-regular; (5) a separated, Cauchy, zero space which is not weakly regular; (6) a topological space which is regular and $\mu $-regular but not $z$-regular; (7) a filter, zero space which is regular and $z$-regular but not completely regular; and, (8) a regular Hausdorff topological space which is $z$-regular but not completely regular.},
author = {Bentley, H. L., Lowen-Colebunders, Eva},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {merotopic space; nearness space; Cauchy space; filter merotopic space; pretopological space; zero space; complete regularity; weak regularity; $z$-regularity; $\mu $-regularity; -regularity; -regularity; filter merotopic space; pretopological space; weak regularity; complete regularity; merotopic spaces; zero space; Cauchy space},
language = {eng},
number = {1},
pages = {129-153},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Completely regular spaces},
url = {http://eudml.org/doc/247257},
volume = {32},
year = {1991},
}

TY - JOUR
AU - Bentley, H. L.
AU - Lowen-Colebunders, Eva
TI - Completely regular spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1991
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 32
IS - 1
SP - 129
EP - 153
AB - We conduct an investigation of the relationships which exist between various generalizations of complete regularity in the setting of merotopic spaces, with particular attention to filter spaces such as Cauchy spaces and convergence spaces. Our primary contribution consists in the presentation of several counterexamples establishing the divergence of various such generalizations of complete regularity. We give examples of: (1) a contigual zero space which is not weakly regular and is not a Cauchy space; (2) a separated filter space which is a $z$-regular space but not a nearness space; (3) a separated, Cauchy, zero space which is $z$-regular but not regular; (4) a separated, Cauchy, zero space which is $\mu $-regular but not regular and not $z$-regular; (5) a separated, Cauchy, zero space which is not weakly regular; (6) a topological space which is regular and $\mu $-regular but not $z$-regular; (7) a filter, zero space which is regular and $z$-regular but not completely regular; and, (8) a regular Hausdorff topological space which is $z$-regular but not completely regular.
LA - eng
KW - merotopic space; nearness space; Cauchy space; filter merotopic space; pretopological space; zero space; complete regularity; weak regularity; $z$-regularity; $\mu $-regularity; -regularity; -regularity; filter merotopic space; pretopological space; weak regularity; complete regularity; merotopic spaces; zero space; Cauchy space
UR - http://eudml.org/doc/247257
ER -

References

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