On a topological choice principle by Murray Bell

In ZF set theory, we investigate the deductive strength of Murray Bellʼs principle (C):For every set {Ai:i∈I} of non-empty sets, there exists a set {Ti:i∈I} such that for every i∈I, Ti is a compact T2 topology on Ai, with regard to various choice forms.Among other results, we prove the following:(1)The Axiom of Multiple Choice (MC) does not imply statement (C) in ZFA set theory.(2)If κ is an infinite well-ordered cardinal number, then (C) + “Every filter base on κ can be extended to an ultrafilter” implies “For every family A={Ai:i∈κ} such that for all i∈κ, |Ai|⩾2, there is a function (called a Kinna–Wagner function) f with domain A such that for all A∈A, ∅≠f(A)⊊A” and “For every natural number n⩾2, every family A={Ai:i∈κ} of non-empty sets each of which has at most n elements has a choice function”.(3)If κ is an infinite well-ordered cardinal number, then (C) + “There exists a free ultrafilter on κ” implies “For every family A={Ai:i∈κ} such that for all i∈κ, |Ai|⩾2, there is an infinite subset B⊆A with a Kinna–Wagner function” and “For every natural number n⩾2, every family A={Ai:i∈κ} of non-empty sets each of which has at most n elements has an infinite subfamily with a choice function”.(4)(C) + “Every compact T2 space is effectively normal” implies MC restricted to families of non-empty sets each expressible as a countable union of finite sets, and “For every family A={Ai:i∈ω} such that for all i∈ω, 2⩽|Ai|<ℵ0, there is an infinite subset B⊆A with a Kinna–Wagner function”.(5)(C) + “For every set X, every countable filter base on X can be extended to an ultrafilter on X” implies ACℵ0, i.e., the axiom of choice for countable families of non-empty sets.(6)(C) restricted to countable families of non-empty sets + “For every set X, every countable filter base on X can be extended to an ultrafilter on X” is equivalent to ACℵ0 + “There exists a free ultrafilter on ω”.(7)(C) restricted to countable families of non-empty sets + “For every set X, every countable filter base on X can be extended to an ultrafilter on X” implies the statements: “The Tychonoff product of a countable family of compact spaces is compact” and “For every infinite set X, the (generalized) Cantor cube 2X is countably compact”.(8)(C) restricted to countable families of non-empty sets does not imply “There exists a free ultrafilter on ω” in ZF.(9)(C) + “The axiom of choice for countable families of non-empty sets of reals” implies “There exists a non-Lebesgue-measurable set of reals”.(10)The conjunction of the Countable Union Theorem (the union of a countable family of countable sets is countable) and “Every infinite set is Dedekind-infinite” does not imply (C) restricted to countable families of non-empty sets, in ZFA set theory.