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 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 ZF 0 (i.e., ZF minus the Axiom of Regularity). 4. The statement: For every set I, every ℵ 0 - sized family of non-empty finite subsets of I has a choice function implies the Tychonoff product of ℵ 0 many non-empty finite discrete subsets of I is compact , is not provable in ZF 0 .