Remarks on the Stone Spaces of the Integers and the Reals without AC

• Autorzy: Herrlich, H. , Keremedis, K. , Tachtsis, E.
• Języki publikacji: EN

Abstrakty

EN

In ZF, i.e., the Zermelo–Fraenkel set theory minus the Axiom of Choice AC, we investigate the relationship between the Tychonoff product 2^P(X), where 2 is 2 = f0; 1g with the discrete topology, and the Stone space S(X) of the Boolean algebra of all subsets of X, where X =ω,R. We also study the possible placement of well-known topological statements which concern the cited spaces in the hierarchy of weak choice principles.

Słowa kluczowe

EN

axiom of choice; weak axioms of choice; compactness; filters; ultrafilters; Stone space; Boolean prime ideal theorem; Loeb spaces

• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2011
• Tom: Vol. 59, no 2
• Strony: 101--114
• Opis fizyczny: Bibliogr. 12 poz.

Twórcy

autor

Herrlich, H.

• Feldhäuser Str. 69 28865 Lilienthal, Germany,

autor

Keremedis, K.

• Department of Mathematics University of the Aegean Karlovassi, Samos 83200, Greece,

autor

Tachtsis, E.

• Department of Statistics and Actuarial-Financial Mathematics University of the Aegean Karlovassi, Samos 83200, Greece,

Bibliografia

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• [2] R. Engelking, General Topology, PWN–Polish Sci. Publ., Warszawa, 1977.
• [3] J. Fossy and M. Morillon, The Baire category property and some notions of compactness, J. London Math. Soc. 57 (1998), 1–19.
• [4] P. Howard and J. E. Rubin, Consequences of the Axiom of Choice, Math. Surveys Monogr. 59, Amer. Math. Soc., Providence, RI, 1998 (http://consequences.emich.edu/conseq.htm).
• [5] K. Keremedis, The compactness of 2R and the axiom of choice, Math. Logic Quart.46 (2000), 569–571.
• [6] —, Tychonoff products of two-element sets and some weakenings of the Boolean prime ideal theorem, Bull. Polish Acad. Sci. Math. 53 (2005), 349–359.
• [7] —, The Boolean prime ideal theorem and products of cofinite topologies, submitted.
• [8] —, Compact and Loeb Hausdorff spaces in ZF and the axiom of choice for families of finite sets, submitted.
• [9] K. Keremedis, E. Felouzis and E. Tachtsis, On the compactness and countable compactness of 2R in ZF, Bull. Polish Acad. Sci. Math. 55 (2007), 293–302.
• [10] P. A. Loeb, A new proof of the Tychonoff theorem, Amer. Math. Monthly 72 (1965), 711–717.
• [11] J. Novák, On the Cartesian product of two compact spaces, Fund. Math. 40 (1953), 106–112.
• [12] E. Tachtsis, On the set-theoretic strength of countable compactness of the Tychonoff product 2R, Bull. Polish Acad. Sci. Math. 58 (2010), 91–107.

Identyfikatory

DOI

10.4064/ba59-2-1