On the extensibility of closed filters in T 1 spaces and the existence of well orderable filter bases

Kyriakos Keremedis; Eleftherios Tachtsis

Commentationes Mathematicae Universitatis Carolinae (1999)

  • Volume: 40, Issue: 2, page 343-353
  • ISSN: 0010-2628

Abstract

top
We show that the statement CCFC = “the character of a maximal free filter F of closed sets in a T 1 space ( X , T ) is not countable” is equivalent to the Countable Multiple Choice Axiom CMC and, the axiom of choice AC is equivalent to the statement CFE 0 = “closed filters in a T 0 space ( X , T ) extend to maximal closed filters”. We also show that AC is equivalent to each of the assertions: “every closed filter ℱ in a T 1 space ( X , T ) extends to a maximal closed filter with a well orderable filter base”, “for every set A ≠ ∅ , every filter ℱ ⊆ 𝒫 ( A ) extends to an ultrafilter with a well orderable filter base” and “every open filter ℱ in a T 1 space ( X , T ) extends to a maximal open filter with a well orderable filter base”.

How to cite

top

Keremedis, Kyriakos, and Tachtsis, Eleftherios. "On the extensibility of closed filters in T$_1$ spaces and the existence of well orderable filter bases." Commentationes Mathematicae Universitatis Carolinae 40.2 (1999): 343-353. <http://eudml.org/doc/248403>.

@article{Keremedis1999,
abstract = {We show that the statement CCFC = “the character of a maximal free filter $F$ of closed sets in a $T_1$ space $(X,T)$ is not countable” is equivalent to the Countable Multiple Choice Axiom CMC and, the axiom of choice AC is equivalent to the statement CFE$_0$ = “closed filters in a $T_0$ space $(X,T)$ extend to maximal closed filters”. We also show that AC is equivalent to each of the assertions: “every closed filter $\mathcal \{F\}$ in a $T_1$ space $(X,T)$ extends to a maximal closed filter with a well orderable filter base”, “for every set $A\ne \emptyset $, every filter $\mathcal \{F\} \subseteq \mathcal \{P\}(A)$ extends to an ultrafilter with a well orderable filter base” and “every open filter $\mathcal \{F\}$ in a $T_1$ space $(X,T)$ extends to a maximal open filter with a well orderable filter base”.},
author = {Keremedis, Kyriakos, Tachtsis, Eleftherios},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {closed filters; bases for filters; characters of filters; ultrafilters; closed filters; bases for filters; characters of filters; ultrafilters; countable multiple choice; axiom of choice},
language = {eng},
number = {2},
pages = {343-353},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On the extensibility of closed filters in T$_1$ spaces and the existence of well orderable filter bases},
url = {http://eudml.org/doc/248403},
volume = {40},
year = {1999},
}

TY - JOUR
AU - Keremedis, Kyriakos
AU - Tachtsis, Eleftherios
TI - On the extensibility of closed filters in T$_1$ spaces and the existence of well orderable filter bases
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1999
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 40
IS - 2
SP - 343
EP - 353
AB - We show that the statement CCFC = “the character of a maximal free filter $F$ of closed sets in a $T_1$ space $(X,T)$ is not countable” is equivalent to the Countable Multiple Choice Axiom CMC and, the axiom of choice AC is equivalent to the statement CFE$_0$ = “closed filters in a $T_0$ space $(X,T)$ extend to maximal closed filters”. We also show that AC is equivalent to each of the assertions: “every closed filter $\mathcal {F}$ in a $T_1$ space $(X,T)$ extends to a maximal closed filter with a well orderable filter base”, “for every set $A\ne \emptyset $, every filter $\mathcal {F} \subseteq \mathcal {P}(A)$ extends to an ultrafilter with a well orderable filter base” and “every open filter $\mathcal {F}$ in a $T_1$ space $(X,T)$ extends to a maximal open filter with a well orderable filter base”.
LA - eng
KW - closed filters; bases for filters; characters of filters; ultrafilters; closed filters; bases for filters; characters of filters; ultrafilters; countable multiple choice; axiom of choice
UR - http://eudml.org/doc/248403
ER -

References

top
  1. Blass A., A model without ultrafilters, Bull. Acad. Sci. Polon., Ser. Sci. Math. Astr. Phys. 25 (1977), 329-331. (1977) Zbl0365.02054MR0476510
  2. Blass A., Prime Ideals yield almost maximal Ideals, Fund. Math. 127 (1986), 56-66. (1986) MR0883153
  3. Brunner N., σ -kompakte raume, Manuscripta Math. 38 (1982), 375-379. (1982) MR0667922
  4. Howard P., Keremedis K., Rubin H., Rubin J.E., Versions of normality and some weak forms of the axiom of choice, to appear in Math. Logic Quarterly 44 (1998). Zbl0911.03027MR1645498
  5. Howard P., Rubin J.E., Consequences of the Axiom of Choice, AMS, Mathematical Surveys and Monographs 59, 1998. Zbl0947.03001MR1637107
  6. Herrlich H., Steprāns J., Maximal filters, continuity, and choice principles, Quaestiones Math. 20 (1997). (1997) MR1625478
  7. Jech T.J., The Axiom of Choice, North-Holland, Amsterdam, 1973. Zbl0259.02052MR0396271
  8. Keremedis K., Disasters in topology without the axiom of choice, preprint, 1996. Zbl1027.03040MR1867681
  9. Levy A., Axioms of multiple choice, Fund. Math. 50 (1962), 475-483. (1962) Zbl0134.24805MR0139528
  10. Munkres J.R., Topology : A first course, Prentice-Hall, Englewood Cliffs NJ, 1975. Zbl0306.54001MR0464128
  11. Rubin H., Rubin J.E., Equivalents of the Axiom of Choice, II, North-Holland, 1985. MR0798475
  12. Herrlich H., Compactness and the Axiom of Choice, Applied Categorical Structures 4 (1996), 1-14. (1996) Zbl0881.54027MR1393958

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.