Compact and Loeb Hausdorff spaces in documentclass{article}usepackage{amssymb}begin{document}pagestyle{empty}$mathsf {ZF}$end{document} and the axiom of choice for families of finite sets

Given a set X, $mathsf {AC}^{mathrm{fin}(X)}$ denotes the statement: “$[X]^{has a choice set” and $mathcal {C}_mathrm{R}big (mathbf {2}^{X}big )$ denotes the family of all closed subsets of the topological space $mathbf {2}^{X}$ whose definition depends on a finite subset of X. We study the interrelations between the statements $mathsf {AC}^{mathrm{fin}(X)},$ $mathsf {AC}^{mathrm{fin}([X]^{has a choice set”. We show:

• (i) $mathsf {AC}^{mathrm{fin}(X)}$ iff $mathsf {AC}^{mathrm{fin}([X]^{
• (ii) $mathsf {AC}_{mathrm{fin}}$ ($mathsf {AC}$ restricted to families of finite sets) iff for every set X, $mathcal {C}_mathrm{R}big (mathbf {2}^{X}big )backslash lbrace varnothing rbrace$ has a choice set.
• (iii) $mathsf {AC}_{mathrm{fin}}$ does not imply “$mathcal {K}big (mathbf {2}^{X}big )backslash lbrace varnothing rbrace$ has a choice set($mathcal {K}(mathbf {X})$ is the family of all closed subsets of the space $mathbf {X}$)
• (iv) $mathcal {K}(mathbf {2}^{X})backslash lbrace varnothing rbrace$ implies $mathsf {AC}^{mathrm{fin}(mathcal {wp }(X))}$ but $mathsf {AC}^{mathrm{fin}(X)}$ does not imply $mathsf {AC}^{mathrm{fin}(mathcal {wp }(X))}$.

We also show that “For every setX, “$mathcal {K}big (mathbf {2}^{X}big )backslash lbrace varnothing rbrace$has a choice set” iff “for every setX, $mathcal {K}big (mathbf {[0,1]}^{X}big )backslash lbrace varnothing rbrace$has a choice set” iff “for every product$mathbf {X}$of finite discrete spaces,$mathcal {K}(mathbf {X})backslash lbrace varnothing rbrace$ has a choice set”.