Choice principles in elementary topology and analysis

Horst Herrlich

Commentationes Mathematicae Universitatis Carolinae (1997)

  • Volume: 38, Issue: 3, page 545-552
  • ISSN: 0010-2628

Abstract

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Many fundamental mathematical results fail in ZF, i.e., in Zermelo-Fraenkel set theory without the Axiom of Choice. This article surveys results — old and new — that specify how much “choice” is needed precisely to validate each of certain basic analytical and topological results.

How to cite

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Herrlich, Horst. "Choice principles in elementary topology and analysis." Commentationes Mathematicae Universitatis Carolinae 38.3 (1997): 545-552. <http://eudml.org/doc/248085>.

@article{Herrlich1997,
abstract = {Many fundamental mathematical results fail in ZF, i.e., in Zermelo-Fraenkel set theory without the Axiom of Choice. This article surveys results — old and new — that specify how much “choice” is needed precisely to validate each of certain basic analytical and topological results.},
author = {Herrlich, Horst},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Axiom of (Countable) Choice; Boolean Prime Ideal Theorem; Theorems of Ascoli; Baire; Čech-Stone and Tychonoff; compact; Lindelöf and orderable spaces; axiom of choice; Zermelo-Fraenkel set theory; compact space; Lindelöf space; orderable space},
language = {eng},
number = {3},
pages = {545-552},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Choice principles in elementary topology and analysis},
url = {http://eudml.org/doc/248085},
volume = {38},
year = {1997},
}

TY - JOUR
AU - Herrlich, Horst
TI - Choice principles in elementary topology and analysis
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1997
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 38
IS - 3
SP - 545
EP - 552
AB - Many fundamental mathematical results fail in ZF, i.e., in Zermelo-Fraenkel set theory without the Axiom of Choice. This article surveys results — old and new — that specify how much “choice” is needed precisely to validate each of certain basic analytical and topological results.
LA - eng
KW - Axiom of (Countable) Choice; Boolean Prime Ideal Theorem; Theorems of Ascoli; Baire; Čech-Stone and Tychonoff; compact; Lindelöf and orderable spaces; axiom of choice; Zermelo-Fraenkel set theory; compact space; Lindelöf space; orderable space
UR - http://eudml.org/doc/248085
ER -

References

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  1. Alas O.T., The Axiom of choice and two particular forms of Tychonoff theorem, Portugal. Math. 28 (1968), 75-76. (1968) MR0281600
  2. Banaschewski B., Compactification and the axiom of choice, unpublished manuscript, 1979. 
  3. Bentley H.L., Herrlich H., Compactness and rings of continuous functions - without the axiom of choice, to appear. Zbl0986.54029MR1722566
  4. Bentley H.L., Herrlich H., Countable choice and pseudometric spaces, to appear. Zbl0922.03068MR1617460
  5. Blass A., A model without ultrafilters, Bull. Acad. Sci. Polon., Sér. Sci. Math. Astr. Phys. 25 (1977), 329-331. (1977) Zbl0365.02054MR0476510
  6. Comfort W.W., A theorem of Stone-Čech type, and a theorem of Tychonoff type, without the axiom of choice; and their realcompact analogues, Fund. Math. 63 (1988), 97-110. (1988) MR0236880
  7. Good C., Tree I.J., Continuing horrors of topology without choice, Topol. Appl. 63 (1995), 79-90. (1995) Zbl0822.54001MR1328621
  8. Goodstein R.L., Existence in Mathematics, in: Logic and Foundations of Mathematics (eds. D. van Dalen et al.), Wolters-Noordhoff Publ. Co., 1968, pp.70-82. Zbl0162.30901MR0247998
  9. Halpern J.D., Lévy A., The Boolean prime ideal theorem does not imply the axiom of choice, Proc. of Symposium Pure Math. of the AMS 13 (1971), Part I, 83-134. (1971) MR0284328
  10. Herrlich H., Compactness and the Axiom of Choice, Appl. Categ. Structures 4 (1996), 1-14. (1996) Zbl0881.54027MR1393958
  11. Herrlich H., An effective construction of a free ultrafilter, Papers on Gen. Topol. Appl. (eds. S. Andima et al.), Annals New York Acad Sci. 806 (1996), 201-206. (1996) MR1429654
  12. Herrlich H., The Ascoli Theorem is equivalent to the Boolean Prime Ideal Theorem, to appear. Zbl0880.54005MR1602169
  13. Herrlich H., The Ascoli Theorem is equivalent to the Axiom of Choice, to appear. 
  14. Herrlich H., Steprāns J., Maximal filters, continuity and choice principles, to appear in Quaestiones Math. 20 (1997). MR1625478
  15. Herrlich H., Strecker G.E., When in ℕ Lindelöf?, to appear in Comment. Math. Univ. Carolinae 38 (1997). MR1485075
  16. Hilbert D., Über das Unendliche, Mathem. Annalen 95 (1926), 161-190. (1926) MR1512272
  17. Jaegermann M., The Axiom of Choice and two definitions of continuity, Bull. Acad. Polon. Sci, Sér. Sci, Math., Astr. et Phys. 13 (1965), 699-704. (1965) Zbl0252.02059MR0195711
  18. Jech T., Eine Bemerkung zum Auswahlaxiom, Časopis Pěst. Mat. 93 (1968), 30-31. (1968) Zbl0167.27402MR0233706
  19. Jech T.J., The Axiom of Choice, North Holland, Amsterdam, 1973. Zbl0259.02052MR0396271
  20. Jensen R.B., Independence of the Axiom of Dependent Choices from the Countable Axiom of Choice, J. Symb. Logic 31 (1966), 294. (1966) 
  21. Kelley J.L., The Tychonoff product theorem implies the axiom of choice, Fund. Math. 37 (1950), 75-76. (1950) Zbl0039.28202MR0039982
  22. Loś J., Ryll-Nardzewski C., Effectiveness of the representation theory for Boolean algebras, Fund. Math. 41 (1955), 49-56. (1955) MR0065527
  23. Maddy P., Believing the axioms I., J. Symb. Logic 53 (1988), 481-511. (1988) Zbl0652.03033MR0947855
  24. Moore G.H., Zermelo's Axiom of Choice. Its Origins, Developments and Influence, Springer, New York, 1982. MR0679315
  25. Mycielski J., Two remarks on Tychonoff's product theorem, Bull. Acad. Polon. Sci. Sér. Sci. Math., Astr. Phys. 12 (1964), 439-441. (1964) Zbl0138.17703MR0215731
  26. Pincus D., Adding Dependent Choice to the Boolean Prime Ideal Theorem, Logic Colloq. 76 (1977), 547-565. (1977) MR0480027
  27. Rubin H., Rubin J.E., Equivalents of the Axiom of Choice II., North Holland, Amsterdam, 1985. MR0798475
  28. Rubin H., Scott D., Some topological theorems equivalent to the Boolean prime ideal theorem, Bull. Amer. Math. Soc. 60 (1954), 389. (1954) 
  29. Sierpiñski W., Sur le rôle de l'axiome de M. Zermelo dans l'Analyse moderne, Compt. Rendus Hebdomadaires des Sēances de l'Academie des Sciences, Paris 193 (1916), 688-691. (1916) 
  30. Sierpiñski W., L'axiome de M. Zermelo et son rôle dans la théorie des ensembles et l'analyse, Bull. Acad. Sci. Cracovie, Cl. Sci. Math., Sér. A (1918), 97-152. (1918) 
  31. Douwen E.K., Horrors of topology without AC: a nonnormal orderable space, Proc. Amer. Math. Soc. 95 (1985), 101-105. (1985) Zbl0574.03039MR0796455
  32. Ward L.E., A weak Tychonoff theorem and the axiom of choice, Proc. Amer. Math. Soc. 13 (1962), 757-758. (1962) Zbl0112.14301MR0186537

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