Compactness and the axiom of choice

Applied Categorical Structures

Volumn 4, Issue 1, 1996, Pages 1-14

  Herrlich, Horst ^a  

^a UNIVERSITY OF BREMEN   (Germany)

Author keywords

Axiom of choice; Boolean prime ideal theorem; Compact space; Epireflective subcategory; Free ultrafilter; Topological product

Indexed keywords

EID: 0002353701     PISSN: 09272852     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF00124110     Document Type: Article

References (37)

1. Alas, O. T.: The axiom of choice and two particular forms of Tychonoff theorem, Portugal. Math. 28 (1969), 75-76.
2. Alexandroff, P. and Urysohn, P.: Mémoire sur les espaces topologiques compacts, Verh. Nederl. Akad. Wetensch. Aft. Naturk. Sect. I 14 (1929), 1-96.
3. Banaschewski, B.: Compactification and the axiom of choice, Unpublished manuscript, 1979.
4. Bentley, H. L. and Herrlich, H.: Compactness and rings of continuous functions - without the axiom of choice, To appear (1995+).
5. Blass, A.: A model without ulrafilters, Bull. Acad. Sci. Polon., Sér. Sci. Math. Astr. Phvs. 25 (1977), 329-331.
6. Čech, E.: On bicompact spaces, Ann. Math. 38 (1937), 823-844.
7. Chandler, R. E.: An alternative construction of βX and υX, Proc. Amer. Math. Soc. 32 (1972), 315-318.
8. 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 (1968), 97-110.
9. Feferman, S.: Some applications of the notion of forcing and generic sets, Fund. Math. 56 (1965), 325-345.
10. Gillman, L. and Jerison, M.: Rings of Continuous Functions, Van Nostrand, 1960.
11. Halpern, J. D.: The independence of the axiom of choice from the Boolean prime ideal theorem, Fund. Math. 55 (1964), 57-66.
12. Halpern, J. D. and 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.
13. Hartogs, F.: Über das Problem der Wohlordnung, Math. Annalen 76 (1915), 138-143.
14. Herrlich, H.: Wann sind alle stetigen Abbildungen in Y konstant? Math. Zeitschr. 90 (1965), 152-154.
15. Herrlich, H.: C-kompakte Räume, Math. Zeitschr. 96 (1967), 228-255.
16. Jech, T. J.: The Axiom of Choice, North-Holland, Amsterdam, 1973.
17. Kelley, J. L.: The Tychonoff product theorem implies the axiom of choise, Fund. Math. 37 (1950), 75-76.
18. Kennison, J. F.: Reflective functors in general topology and elsewhere, Trans. Amer. Math. Soc. 118 (1965), 303-315.
19. Krull, W.: Die Idealtheorie in Ringen ohne Endlichkeitsbedingungen, Mathem. Annalen 101 (1929), 729-744.
20. Läuchli, H.: Auswahlaxiom in der Algebra, Commentarii Math. Helvetici 37 (1962-63), 1-18.
21. Łoś, J. and Ryll-Nardzewski, C.: On the application of Tychonoff's theorem in mathematical proofs, Fund. Math. 38 (1951), 233-237.
22. Łoś, J. and Ryll-Nardzewski, C.: Effectiveness of the representation theory for Boolean algebras, Fund. Math. 41 (1955), 49-56.
23. Moore, G. H.: Zermelo's Axiom of Choice. Its Origins, Developments and Influence, Springer, New York, 1982.
24. Mrówka, S.: Compactness and product spaces, Colloq. Math. 7 (1959), 19-22.
25. Mycielski, J.: Two remarks on Tychonoff's product theorem, Bull. Acad. Polon. Sci. Sér. Sci. Math., Astr. Phys. 12 (1964), 439-441.
26. Rubin, H. and Rubin, J. E.: Equivalents of the Axiom of Choice II, North Holland, Amsterdam, 1985.
27. Rubin, H. and Scott, D.: Some topological theorems equivalent to the Boolean prime ideal theorem, Bull. Amer. Math. Soc. 60 (1954), 389.
28. Salbany, S.: On compact* spaces and compactifications, Proc. Amer. Math. Soc. 45 (1974), 274-280.
29. Sierpiński, W.: L'axiome de M. Zermelo et son rôle dans la théorie des ensembles et l'analyse, Bull. l'Acad. Sci. Cracovie Cl. Sci. Math. Sér. A (1918), 97-152.
30. Solovay, R. M.: A model of set theory in which all sets of reals are Lebesgue measurable, Ann. Math. 92 (1970), 1-56.
31. Stone, M. H.: The theory of representations for Boolean algebras, Trans. Amer. Math. Soc. 40 (1936), 37-111.
32. Stone, M. H.: Applications of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc. 41 (1937), 375-481.
33. Tarski, A.: Ein Überdeckungssatz für endliche Mengen, Fund. Math. 30 (1938), 156-163.
34. Tychonoff, S.: Über die topologische Erweiterung von Räumen, Mathem. Annalen 105 (1930), 544-561.
35. Tychonoff, S.: Ein Fixpunktsatz, Mathem. Annalen 111 (1935), 767-776.
36. Ward, L. E.: A weak Tychonoff theorem and the axiom of choice, Proc. Amer. Math. Soc. 13 (1962), 757-758.
37. Zermelo, E.: Beweis, daß jede Menge wohlgeordnet werden kann, Mathem. Annalen 59 (1904), 514-516.