On a topological choice principle by Murray Bell

In ZF set theory, we investigate the deductive strength of Murray Bellʼs principle (C): ery set { A i : i ∈ I } of non-empty sets, there exists a set { T i : i ∈ I } such that for every i ∈ I , T i is a compact T 2 topology on A i , regard to various choice forms. other results, we prove the following:(1) iom of Multiple Choice (MC) does not imply statement (C) in ZFA set theory. s an infinite well-ordered cardinal number, then (C) + “Every filter base on κ can be extended to an ultrafilter” implies “For every family A = { A i : i ∈ κ } such that for all i ∈ κ , | A i | ⩾ 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 = { A i : i ∈ κ } of non-empty sets each of which has at most n elements has a choice function”. s an infinite well-ordered cardinal number, then (C) + “There exists a free ultrafilter on κ” implies “For every family A = { A i : i ∈ κ } such that for all i ∈ κ , | A i | ⩾ 2 , there is an infinite subset B ⊆ A with a Kinna–Wagner function” and “For every natural number n ⩾ 2 , every family A = { A i : i ∈ κ } of non-empty sets each of which has at most n elements has an infinite subfamily with a choice function”. “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 = { A i : i ∈ ω } such that for all i ∈ ω , 2 ⩽ | A i | < ℵ 0 , there is an infinite subset B ⊆ A with a Kinna–Wagner function”. “For every set X, every countable filter base on X can be extended to an ultrafilter on X” implies AC ℵ 0 , i.e., the axiom of choice for countable families of non-empty sets. stricted 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 AC ℵ 0 + “There exists a free ultrafilter on ω”. stricted 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 2 X is countably compact”. stricted to countable families of non-empty sets does not imply “There exists a free ultrafilter on ω” in ZF. “The axiom of choice for countable families of non-empty sets of reals” implies “There exists a non-Lebesgue-measurable set of reals”. njunction 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.