Compact and Loeb Hausdorff spaces in ZF and the axiom of choice for families of finite sets

摘要

Given a set X, AC~(fin(X)) denotes the statement: “[X]~(<ω) {O} has a choice set” and C_R (2~X) denotes the family of all closed subsets of the topological space 2~X whose definition depends on a finite subset of X.We study the interrelations between the statements AC~(fin(X)) , AC~(fin([X]< ω) ) , AC~(fin(F_n (X ,2))) , AC~(fin(φ(X ))) and “C_R (2~X){O} has a choice set”. We show: (i) AC~(fin(X)) iff AC~(fin([X]< ω) ) iff C_R (2~X){O} has a choice set iff AC~(fin(F_n (X ,2))) . (ii) AC_(fin) (AC restricted to families of finite sets) iff for every set X, C_R (2~X){O} has a choice set. (iii) AC_(fin) does not imply “K(2~X){O} has a choice set(K(X) is the family of all closed subsets of the space X) (iv) K(2~X){O} implies AC~(fin(φ(X))) but AC~(fin(X)) does not imply AC~(fin(φ(X))) . We also show that “For every set X, “K(2~X){O} has a choice set” iff “for every set X, K([0, 1]~X){O} has a choice set” iff “for every product X of finite discrete spaces, K(X){O} has a choice set”.