On BPI Restricted to Boolean Algebras of Size Continuum

• Autorzy: Hall E. , Keremedis K.

Identyfikatory

DOI

10.4064/ba61-1-2

• Języki publikacji: EN

Abstrakty

EN

(i) The statement P(ω)=“every partition of R has size ≤|R|” is equivalent to the proposition R(ω)=“for every subspace Y of the Tychonoff product 2P(ω) the restriction B|Y={Y∩B:B∈B} of the standard clopen base B of 2P(ω) to Y has size ≤|P(ω)|”. (ii) In ZF, P(ω) does not imply “every partition of P(ω) has a choice set”. (iii) Under P(ω) the following two statements are equivalent: (a) For every Boolean algebra of size ≤|R| every filter can be extended to an ultrafilter. (b) Every Boolean algebra of size ≤|R| has an ultrafilter.

Słowa kluczowe

EN

weak axioms of choice; filters; ultrafilters; Boolean prime ideal theorem; regular open set

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2013
• Tom: Vol. 61, no 1
• Strony: 9--21
• Opis fizyczny: Bibliogr. 10 poz.

Twórcy

autor

Hall E.

• Department of Mathematics & Statistics College of Arts & Sciences University of Missouri – Kansas City 206 Haag Hall, 5100 Rockhill Rd Kansas City, MO 64110, USA

autor

Keremedis K.

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

Bibliografia

• [1] A. Blass, The model of set theory generated by countably many generic reals, J. Symbolic Logic 46 (1981), 732–752.
• [2] E. Hall and K. Keremedis, Independent families and some notions of finiteness, submitted.
• [3] E. Hall and K. Keremedis, Cech–Stone compactifications of discrete spaces in ZF and some weak forms of the Boolean prime ideal theorem, Topology Proc. 41 (2013), 111–122.
• [4] H. Herrlich, K. Keremedis and E. Tachtsis, Remarks on the Stone spaces of the integers and the reals without AC, Bull. Polish Acad. Sci. Math. 59 (2011), 101–114.
• [5] E. Hall, K. Keremedis and E. Tachtsis, The existence of free ultrafilters on ! Does not imply the extension of filters on ! to ultrafilters, Math. Logic Quart., to appear.
• [6] P. Howard and J. E. Rubin, Consequences of the Axiom of Choice, Math. Surveys Monogr. 59, Amer. Math. Soc., Providence, RI, 1998.
• [7] T. Jech, The Axiom of Choice, North-Holland, Amsterdam, 1973.
• [8] K. Keremedis, Tychonoff products of two-element sets and some weakenings of the Boolean prime ideal theorem, Bull. Polish Acad. Sci. Math. 53 (2005), 349–359.
• [9] K. Keremedis and E. Tachtsis, Countable compact Hausdorff spaces need not be metrizable in ZF, Proc. Amer. Math. Soc. 135 (2007), 1205–1211.
• [10] G. P. Monro, On generic extensions without the axiom of choice, J. Symbolic Logic 48 (1983), 39–52.