Canonical subbase-compactness of topological products

For topological products the concept of canonical subbase-compactness is introduced, and the question analyzed under what conditions such products are canonically subbase-compact in ZF-set theory. s: (1) Products of finite spaces are canonically subbase-compact iff AC ( fin ) , the axiom of choice for finite sets, holds. oducts of n-element spaces are canonically subbase-compact iff AC ( < n ) , the axiom of choice for sets with less than n elements, holds. oducts of compact spaces are canonically subbase-compact iff AC, the axiom of choice, holds. l powers X I of a compact space X are canonically subbase compact iff X is a Loeb-space. results imply that in ZF the implications compact ⇒ canonically subbase-compact ⇒ subbase-compact are both irreversible.