Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-25T19:08:35.855Z Has data issue: false hasContentIssue false

Definitions of compactness and the axiom of choice

Published online by Cambridge University Press:  12 March 2014

Omar De La Cruz
Affiliation:
Mathematics Department, Purdue University, West Lafayette, IN 47907, USA, E-mail: odlc@math.purdue.edu
Eric Hall
Affiliation:
Mathematics Department, Purdue University, West Lafayette, IN 47907, USA, E-mail: ericeric@math.purdue.edu
Paul Howard
Affiliation:
Mathematics Department, Eastern Michigan University, Yipsilanti, MI 48197, USA, E-mail: phoward@emunix.emich.edu
Jean E. Rubin
Affiliation:
Mathematics Department, Purdue University, West Lafayette, IN 47907, USA, E-mail: jer@math.purdue.edu
Adrienne Stanley
Affiliation:
Mathematics Department, University of Northern Iowa, Cedar Falls, IA 50614, USA, E-mail: stanley@cns.uni.edu

Abstract

We study the relationships between definitions of compactness in topological spaces and the roll the axiom of choice plays in these relationships.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2002

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[al]Alexandroff, P., Über die Metrisation der Kleinen kompakten topologischen Räume, Mathematische Annalen, vol. 92 (1924), pp. 294301.CrossRefGoogle Scholar
[c]Čech, E., On bicompact spaces, Annals of Mathematics, vol. 38 (1937), pp. 823844.CrossRefGoogle Scholar
[co]Comfort, W. W., A theorem of Stone-Čech Type, and a theorem of Tychonoff Type, without the axiom of choice and their real compact analogues, Fundamenta Mathematicae, vol. 43 (1968), pp. 97110.CrossRefGoogle Scholar
[fr]Fréchet, M., Sur quelque points du calcul fonctionnel, Rendiconti del Circolo Matematico di Palermo, vol. 22 (1906), pp. 174.CrossRefGoogle Scholar
[gj]Gillman, L. and Jerrison, M., Rings of continuous functions, Van Nostrand, 1960.CrossRefGoogle Scholar
[he]Herrlich, H., Compactness and the axiom of choice, Applied Categorical Structures, vol. 4 (1996), pp. 114.CrossRefGoogle Scholar
[ho]Howard, P., Definitions of compactness, this Journal, vol. 55 (1990), pp. 645655.Google Scholar
[hr]Howard, P. and Rubin, J. E., Consequences of the Axiom of Choice, Math. Surveys and Monographs, vol. 59, AMS, 1998, http://www.math.purdue.edu/~jer/cgi-bin/conseq.html.CrossRefGoogle Scholar
[j]Jech, T. J., The Axiom of Choice, North-Holland Publ. Co., 1973.Google Scholar
[k]Kelley, J. L., General topology, second ed., Springer-Verlag, 1985.Google Scholar
[m]Monro, G. P., Independence results concerning Dedekind finite sets, Journal of Australian Mathematical Society, vol. 19 (1975), pp. 3546.CrossRefGoogle Scholar
[mo]Moore, G. H., Zermelo's axiom of choice, Springer-Verlag, 1982.CrossRefGoogle Scholar
[p]Parovičenko, I. I., Topological equivalents of the Tihonov theorem, Doklady Akademii Nauk, vol. 10 (1969), pp. 3334.Google Scholar
[ri]Riesz, F., Stetigkeitsbegriff und abstrakte Mengenlehre, Atti del IV Congresso Internazionale dei Matematici (Roma, 6–11 Aprile 1908), vol. II (1908), pp. 1824.Google Scholar
[s]Sierpinski, W., General topology, University of Toronto Press, 1956.Google Scholar
[th]Thron, W. J., Topological structures. Holt, Rinehart and Winston, 1966.Google Scholar
[tr]Truss, J. K., The axiom of choice for linearly ordered families, Fundamenta Mathematicae, vol. 99 (1978), pp. 122139.CrossRefGoogle Scholar
[ty]Tychonoff, A., Über die topologische Erweiterung von Räumen, Mathematische Annalen. vol. 102 (1930), pp. 544561.CrossRefGoogle Scholar
[ur]Urysohn, P., Über die Metrisation der kompakten topologischen Räume, Mathematische Annalen, vol. 92 (1924), pp. 275293.CrossRefGoogle Scholar
[we]Weierstrass, K., Über die analytische Darstellbarkeit sogenannter willkürlicher Functionen reeller Argumente, Sitzungsberichte der Deutschen Akademie der Wissenschaft zu Berlin. Klasse für Mathematik, Physik und Technik, (1885), pp. 633–639, 789805.Google Scholar
[w]Willard, S., General topology, Addison-Wesley Publ. Co., 1968.Google Scholar
[wo]Wolk, E. S., On theorems of Tychonoff, Alexander, and R. Rado, Proceedings of the American Mathematical Society, vol. 18 (1967), pp. 113115.CrossRefGoogle Scholar