Compact and Loeb Hausdorff spaces in equation image and the axiom of choice for families of finite sets

Mathematical Logic Quarterly 58 (3):130-138 (2012)

Abstract
Given a set X, equation image denotes the statement: “equation image has a choice set” and equation image denotes the family of all closed subsets of the topological space equation image whose definition depends on a finite subset of X. We study the interrelations between the statements equation image equation image equation image equation image and “equation imagehas a choice set”. We show: equation image iff equation image iff equation image has a choice set iff equation image. equation image iff for every set X, equation image has a choice set. equation image does not imply “equation image has a choice set equation image implies equation image but equation image does not imply equation image.We also show that “For every setX, “equation imagehas a choice set” iff “for every setX, equation imagehas a choice set” iff “for every productequation imageof finite discrete spaces,equation image has a choice set”.
Keywords weak axioms of choice  Axiom of choice  Boolean prime ideal theorem. msc (201) E325  Loeb spaces  Tychonoff products  54B10  54A35
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DOI 10.1002/malq.201020039
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References found in this work BETA

The Structure of Amorphous Sets.J. K. Truss - 1995 - Annals of Pure and Applied Logic 73 (2):191-233.
Products of Compact Spaces in the Least Permutation Model.Norbert Brunner - 1985 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 31 (25-28):441-448.
Products of Compact Spaces in the Least Permutation Model.Norbert Brunner - 1985 - Mathematical Logic Quarterly 31 (25‐28):441-448.

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