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Products of some special compact spaces and restricted forms of AC

Published online by Cambridge University Press:  12 March 2014

Kyriakos Keremedis  and
Eleftherios Tachtsis
Kyriakos Keremedis
Affiliation:
University of the Aegean, Department of Mathematics, Karlovassi, Samos 83200, Greece. E-mail: [email protected]
Eleftherios Tachtsis
Affiliation:
University of the Aegean, Department of Statistics and Actuarial-Financial Mathematics, Karlovassi, Samos 83200, Greece. E-mail: [email protected]
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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]Iwhich consists offunctions 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).

4. The statement: For every set I, every ℵ0-sized family of non-empty finite subsets of I has a choice function implies the Tychonoff product of ℵ0many non-empty finite discrete subsets of I is compact, is not provable in ZF0.

Keywords

Axiom of Choiceweak Axioms of ChoiceTychonoff productscompactnesspermutation model
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 75 , Issue 3 , September 2010 , pp. 996 - 1006
Copyright
Copyright © Association for Symbolic Logic 2010

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