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 , \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {AC}^{\mathrm{fin}(X)}$\end{document} denotes the statement: “ \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$[X]^{<\omega }\backslash \lbrace \varnothing \rbrace$\end{document} has a choice set ” and \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )$\end{document} denotes the family of all closed subsets of the topological space \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbf {2}^{X}$\end{document} whose definition depends on a finite subset of X . We study the interrelations between the statements \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {AC}^{\mathrm{fin}(X)},$\end{document} \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {AC}^{\mathrm{fin}([X]^{<\omega })},$\end{document} \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {AC}^{\mathrm{fin} (F_{n}(X,2))},$\end{document} \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {AC}^{\mathrm{fin}(\mathcal {\wp }(X))}$\end{document} and “ \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$\end{document} has a choice set ”. We show: (i) \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {AC}^{\mathrm{fin}(X)}$\end{document} iff \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {AC}^{\mathrm{fin}([X]^{<\omega } )}$\end{document} iff \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$\end{document} has a choice set iff \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {AC}^{\mathrm{fin}(F_{n}(X,2))}$\end{document} . (ii) \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {AC}_{\mathrm{fin}}$\end{document} ( \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {AC}$\end{document} restricted to families of finite sets) iff for every set X , \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$\end{document} has a choice set. (iii) \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {AC}_{\mathrm{fin}}$\end{document} does not imply “ \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {K}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$\end{document} has a choice set( \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {K}(\mathbf {X})$\end{document} is the family of all closed subsets of the space \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbf {X}$\end{document} ) (iv) \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {K}(\mathbf {2}^{X})\backslash \lbrace \varnothing \rbrace$\end{document} implies \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {AC}^{\mathrm{fin}(\mathcal {\wp }(X))}$\end{document} but \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {AC}^{\mathrm{fin}(X)}$\end{document} does not imply \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {AC}^{\mathrm{fin}(\mathcal {\wp }(X))}$\end{document} . We also show that “ For every set X , “ \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {K}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$\end{document} has a choice set ” iff “ for every set X , \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {K}\big (\mathbf {[0,1]}^{X}\big )\backslash \lbrace \varnothing \rbrace$\end{document} has a choice set ” iff “ for every product \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbf {X}$\end{document} of finite discrete spaces, \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {K}(\mathbf {X})\backslash \lbrace \varnothing \rbrace$\end{document} has a choice set ”.