Journal of Symbolic Logic

Products of some special compact spaces and restricted forms of AC

Kyriakos Keremedis and Eleftherios Tachtsis

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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.

Article information

Source
J. Symbolic Logic, Volume 75, Issue 3 (2010), 996-1006.

Dates
First available in Project Euclid: 9 July 2010

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1278682212

Digital Object Identifier
doi:10.2178/jsl/1278682212

Mathematical Reviews number (MathSciNet)
MR2723779

Zentralblatt MATH identifier
1208.03051

Subjects
Primary: 03E25: Axiom of choice and related propositions 03E35: Consistency and independence results 54B10: Product spaces 54D30: Compactness

Keywords
Axiom of Choice weak Axioms of Choice Tychonoff products compactness permutation model

Citation

Keremedis, Kyriakos; Tachtsis, Eleftherios. Products of some special compact spaces and restricted forms of AC. J. Symbolic Logic 75 (2010), no. 3, 996--1006. doi:10.2178/jsl/1278682212. https://projecteuclid.org/euclid.jsl/1278682212


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