In ZF set theory, we investigate the deductive strength of Murray Bell's principle (C): For every set {A(i): i is an element of I} of non-empty sets, there exists a set {T-i: i is an element of I} such that for every i is an element of I, T-i is a compact T-2 topology on A(i), 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 kappa is an infinite well-ordered cardinal number, then (C) + "Every filter base on kappa can be extended to an ultrafilter" implies "For every family A = {A(i): i is an element of kappa} such that for all i is an element of kappa, \A(i)\ >= 2, there is a function (called a Kinna-Wagner function) f with domain A such that for all A is an element of A, empty set not equal f (A) not subset of A" and "For every natural number n >= 2, every family A = {A(i): i is an element of kappa} of non-empty sets each of which has at most n elements has a choice function". (3) If kappa is an infinite well-ordered cardinal number, then (C) + "There exists a free ultrafilter on kappa" implies "For every family A = {A(i): i is an element of kappa} such that for all i E K, \A(i)\ >= 2, there is an infinite subset B subset of A with a Kinna-Wagner function" and "For every natural number n >= 2, every family A = {A(i): i is an element of kappa} of non-empty sets each of which has at most n elements has an infinite subfamily with a choice function". (4) (C) + "Every compact T-2 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 is an element of omega} such that for all i is an element of omega, 2 <= \A(i)\ < aleph(0), there is an infinite subset B subset of 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 AC(aleph 0), 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 AC(aleph 0) + "There exists a free ultrafilter on omega". (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 2(X) is countably compact". (8) (C) restricted to countable families of non-empty sets does not imply "There exists a free ultrafilter on omega" 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. (C) 2012 Elsevier B.V. All rights reserved.
