J. Symbolic Logic 75 (3), 996-1006, (September 2010) DOI: 10.2178/jsl/1278682212
Kyriakos Keremedis, Eleftherios Tachtsis
KEYWORDS: axiom of choice, weak Axioms of Choice, Tychonoff products, compactness, permutation model, 03E25, 03E35, 54B10, 54D30
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]ⁱ 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 ℵ₀-sized family of non-empty finite subsets of I has a choice function implies the Tychonoff product of ℵ₀ many non-empty finite discrete subsets of I is compact, is not provable in ZF⁰.