Products of some special compact spaces and restricted forms of AC

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]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.