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

Abstract

AbstractGiven 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: \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}. \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. \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}) \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 setX, “\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 setX, \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”.