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

top
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

top

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

top
  1. Bentley H.L., H. Herrlich H., The reals and the reals, Gen. Topol. Appl. 9 (1978), 221-232. (1978) Zbl0402.54027MR0510903
  2. Bentley H.L., Herrlich H., Completion as reflection, Comment. Math. Univ. Carolinae 19 (1978), 541-568. (1978) Zbl0407.54020MR0508960
  3. Bentley H.L., Herrlich H., Completeness for nearness spaces, In: Topological Structures II, part 1, Math. Centre Tracts 115, Amsterdam, 1979, 29-40. Zbl0463.54022MR0565823
  4. Bentley H.L., Herrlich H., The coreflective hull of the contigual spaces in the category of merotopic spaces, In: Categorical Aspects of Topology and Analysis, Lect. Notes in Math. 915, Springer-Verlag, Berlin, 1982, 16-26. Zbl0561.54019MR0659880
  5. Bentley H.L., Herrlich H., Lowen-Colebunders E., Convergence, J. Pure Appl. Algebra, to appear. Zbl0741.54006MR1082777
  6. Bentley H.L., Herrlich H., Ori R.G., Zero sets and complete regularity for nearness spaces, In: Categorical Topology and its Relations to Analysis, Algebra and Combinatorics, Prague, Czechoslovakia, 22-27 August 1988, Ed. Jiří Adámek and Saunders MacLane, World Scientific Publ. Co., Singapore, 1989, 446-461. MR1047917
  7. Bentley H.L., Herrlich H., Robertson W.A., Convenient categories for topologists, Comment. Math. Univ. Carolinae 17 (1976), 207-227. (1976) Zbl0327.54001MR0425890
  8. Butzmann H.P., Müller B., Topological 𝒞 -embedded spaces, General Topol. and its Appl. 6 (1976), 17-20. (1976) MR0388309
  9. Čech E., Topological Spaces, Revised edition by Z. Frolík and M. Katětov, Publ. House of the Czechoslovak Acad. of Sci., Prague, and Interscience Publ., London, 1966. MR0211373
  10. Choquet G., Convergences, Ann. Univ. Grenoble Sect. Sci. Math. Phys. (NS) 23 (1948), 57-112. (1948) Zbl0031.28101MR0025716
  11. Engelking R., General Topology, Heldermann Verlag, Berlin, 1989. Zbl0684.54001MR1039321
  12. Fisher H.R., Limesräume, Math. Ann. 137 (1959), 269-303. (1959) MR0109339
  13. Henriksen M., Johnson D.G., On the structure of a class of archimedean lattice-ordered algebras, Fund. Math. 50 (1961), 73-94. (1961) Zbl0099.10101MR0133698
  14. Herrlich H., A concept of nearness, Gen. Topol. Appl. 4 (1974), 191-212. (1974) Zbl0288.54004MR0350701
  15. Herrlich H., Topological structures, In: Topological Structures I, Math. Centre Tracts 52, 1974, 59-122. Zbl0589.54018MR0358706
  16. Herrlich H., Categorical topology 1971-1981, In: General Topology and its Relations to Modern Analysis and Algebra V, Proceedings of the Fifth Prague Topological Symposium 1981, Heldermann Verlag, Berlin, 1983, 279-383. Zbl0502.54001MR0698425
  17. Herrlich H., Topologie II: Uniforme Räume, Heldermann Verlag, Berlin, 1988. Zbl0644.54001MR0950372
  18. Katětov M., Allgemeine Stetigkeitsstrukturen, Proc. Intern. Congr. Math. Stockholm 1962 (1963), 473-479. MR0176431
  19. Katětov M., On continuity structures and spaces of mappings, Comment. Math. Univ. Carolinae 6 (1965), 257-278. (1965) MR0193608
  20. Keller H., Die Limes-uniformisierbarkeit der Limesräume, Math. Ann. 176 (1968), 334-341. (1968) Zbl0155.50302MR0225280
  21. Kent D., Richardson G., Regular completions of Cauchy spaces, Pacific J. Math. 51 (1974), 483-490. (1974) Zbl0291.54024MR0390989
  22. Lowen-Colebunders E., Function Classes of Cauchy Continuous Maps, Marcel Dekker, Inc., New York, 1989. Zbl0697.54001MR1001563
  23. Preuß G., Theory of Topological Spaces, D. Reidel Publ. Co., Dordrecht, 1988. 
  24. Robertson W.A., Convergence as a Nearness Concept, Thesis, Carleton University, Ottawa, 1975. 
  25. Šanin N., On separation in topological spaces, Dok. Akad. Nauk SSSR 38 (1943), 110-113. (1943) MR0008688
  26. van Est W.T., Freudenthal H., Trennung durch stetige Funktionen in topologischen Räume, Proc. Kon. Ned Ak. v. Wet., Ser A, 54 (1951), 359-368. (1951) MR0046033
  27. Willard S., General Topology, Addison-Wesley Publ. Co., Reading, 1970. Zbl1052.54001MR0264581

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.