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September 2010 Products of some special compact spaces and restricted forms of AC
Kyriakos Keremedis, Eleftherios Tachtsis
J. Symbolic Logic 75(3): 996-1006 (September 2010). DOI: 10.2178/jsl/1278682212

Abstract

We establish the following results:

1. In ZF (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC), for every set I and for every ordinal number α≥ω, the following statements are equivalent:

a. The Tychonoff product of |α| many non-empty finite discrete subsets of I is compact.

b. The union of |α| many non-empty finite subsets of I is well orderable.

2. The statement: For every infinite set I, every closed subset of the Tychonoff product [0,1] which consists of functions with finite support is compact, is not provable in ZF set theory.

3. The statement: For every set I, the principle of dependent choices relativised to I implies the Tychonoff product of countably many non-empty finite discrete subsets of I is compact, is not provable in ZF⁰ (i.e., ZF minus the Axiom of Regularity).

The statement: For every set I, every ℵ-sized family of non-empty finite subsets of I has a choice function implies the Tychonoff product of ℵ many non-empty finite discrete subsets of I is compact, is not provable in ZF⁰.

Citation

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Kyriakos Keremedis. Eleftherios Tachtsis. "Products of some special compact spaces and restricted forms of AC." J. Symbolic Logic 75 (3) 996 - 1006, September 2010. https://doi.org/10.2178/jsl/1278682212

Information

Published: September 2010
First available in Project Euclid: 9 July 2010

zbMATH: 1208.03051
MathSciNet: MR2723779
Digital Object Identifier: 10.2178/jsl/1278682212

Subjects:
Primary: 03E25, 03E35, 54B10, 54D30

Rights: Copyright © 2010 Association for Symbolic Logic

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Vol.75 • No. 3 • September 2010
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