{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,28]],"date-time":"2025-09-28T15:29:05Z","timestamp":1759073345427},"reference-count":14,"publisher":"Wiley","issue":"3","license":[{"start":{"date-parts":[[2012,4,25]],"date-time":"2012-04-25T00:00:00Z","timestamp":1335312000000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematical Logic Qtrly"],"published-print":{"date-parts":[[2012,5]]},"abstract":"Abstract<\/jats:title>Given a set X<\/jats:italic>, \\documentclass{article}\\usepackage{amssymb}\\begin{document}\\pagestyle{empty}$\\mathsf {AC}^{\\mathrm{fin}(X)}$\\end{document}<\/jats:styled-content> denotes the statement: \u201c\\documentclass{article}\\usepackage{amssymb}\\begin{document}\\pagestyle{empty}$[X]^{<\\omega }\\backslash \\lbrace \\varnothing \\rbrace$\\end{document}<\/jats:styled-content> has a choice set<\/jats:italic>\u201d and \\documentclass{article}\\usepackage{amssymb}\\begin{document}\\pagestyle{empty}$\\mathcal {C}_\\mathrm{R}\\big (\\mathbf {2}^{X}\\big )$\\end{document}<\/jats:styled-content> denotes the family of all closed subsets of the topological space \\documentclass{article}\\usepackage{amssymb}\\begin{document}\\pagestyle{empty}$\\mathbf {2}^{X}$\\end{document}<\/jats:styled-content> whose definition depends on a finite subset of X<\/jats:italic>. We study the interrelations between the statements \\documentclass{article}\\usepackage{amssymb}\\begin{document}\\pagestyle{empty}$\\mathsf {AC}^{\\mathrm{fin}(X)},$\\end{document}<\/jats:styled-content> \\documentclass{article}\\usepackage{amssymb}\\begin{document}\\pagestyle{empty}$\\mathsf {AC}^{\\mathrm{fin}([X]^{<\\omega })},$\\end{document}<\/jats:styled-content> \\documentclass{article}\\usepackage{amssymb}\\begin{document}\\pagestyle{empty}$\\mathsf {AC}^{\\mathrm{fin} (F_{n}(X,2))},$\\end{document}<\/jats:styled-content> \\documentclass{article}\\usepackage{amssymb}\\begin{document}\\pagestyle{empty}$\\mathsf {AC}^{\\mathrm{fin}(\\mathcal {\\wp }(X))}$\\end{document}<\/jats:styled-content> and \u201c\\documentclass{article}\\usepackage{amssymb}\\begin{document}\\pagestyle{empty}$\\mathcal {C}_\\mathrm{R}\\big (\\mathbf {2}^{X}\\big )\\backslash \\lbrace \\varnothing \\rbrace$\\end{document}<\/jats:styled-content>has a choice set<\/jats:italic>\u201d. We show:\n\n\\documentclass{article}\\usepackage{amssymb}\\begin{document}\\pagestyle{empty}$\\mathsf {AC}^{\\mathrm{fin}(X)}$\\end{document}<\/jats:styled-content> iff \\documentclass{article}\\usepackage{amssymb}\\begin{document}\\pagestyle{empty}$\\mathsf {AC}^{\\mathrm{fin}([X]^{<\\omega } )}$\\end{document}<\/jats:styled-content> iff \\documentclass{article}\\usepackage{amssymb}\\begin{document}\\pagestyle{empty}$\\mathcal {C}_\\mathrm{R}\\big (\\mathbf {2}^{X}\\big )\\backslash \\lbrace \\varnothing \\rbrace$\\end{document}<\/jats:styled-content> has a choice set iff \\documentclass{article}\\usepackage{amssymb}\\begin{document}\\pagestyle{empty}$\\mathsf {AC}^{\\mathrm{fin}(F_{n}(X,2))}$\\end{document}<\/jats:styled-content>.<\/jats:p><\/jats:list-item>\n\\documentclass{article}\\usepackage{amssymb}\\begin{document}\\pagestyle{empty}$\\mathsf {AC}_{\\mathrm{fin}}$\\end{document}<\/jats:styled-content> (\\documentclass{article}\\usepackage{amssymb}\\begin{document}\\pagestyle{empty}$\\mathsf {AC}$\\end{document}<\/jats:styled-content> restricted to families of finite sets) iff for every set X<\/jats:italic>, \\documentclass{article}\\usepackage{amssymb}\\begin{document}\\pagestyle{empty}$\\mathcal {C}_\\mathrm{R}\\big (\\mathbf {2}^{X}\\big )\\backslash \\lbrace \\varnothing \\rbrace$\\end{document}<\/jats:styled-content> has a choice set.<\/jats:p><\/jats:list-item>\n\\documentclass{article}\\usepackage{amssymb}\\begin{document}\\pagestyle{empty}$\\mathsf {AC}_{\\mathrm{fin}}$\\end{document}<\/jats:styled-content> does not imply \u201c\\documentclass{article}\\usepackage{amssymb}\\begin{document}\\pagestyle{empty}$\\mathcal {K}\\big (\\mathbf {2}^{X}\\big )\\backslash \\lbrace \\varnothing \\rbrace$\\end{document}<\/jats:styled-content> has a choice set(\\documentclass{article}\\usepackage{amssymb}\\begin{document}\\pagestyle{empty}$\\mathcal {K}(\\mathbf {X})$\\end{document}<\/jats:styled-content> is the family of all closed subsets of the space \\documentclass{article}\\usepackage{amssymb}\\begin{document}\\pagestyle{empty}$\\mathbf {X}$\\end{document}<\/jats:styled-content>)<\/jats:p><\/jats:list-item>\n\\documentclass{article}\\usepackage{amssymb}\\begin{document}\\pagestyle{empty}$\\mathcal {K}(\\mathbf {2}^{X})\\backslash \\lbrace \\varnothing \\rbrace$\\end{document}<\/jats:styled-content> implies \\documentclass{article}\\usepackage{amssymb}\\begin{document}\\pagestyle{empty}$\\mathsf {AC}^{\\mathrm{fin}(\\mathcal {\\wp }(X))}$\\end{document}<\/jats:styled-content> but \\documentclass{article}\\usepackage{amssymb}\\begin{document}\\pagestyle{empty}$\\mathsf {AC}^{\\mathrm{fin}(X)}$\\end{document}<\/jats:styled-content> does not imply \\documentclass{article}\\usepackage{amssymb}\\begin{document}\\pagestyle{empty}$\\mathsf {AC}^{\\mathrm{fin}(\\mathcal {\\wp }(X))}$\\end{document}<\/jats:styled-content>.<\/jats:p><\/jats:list-item>\n<\/jats:list>\nWe also show that \u201cFor every set<\/jats:italic>X<\/jats:italic>, \u201c\\documentclass{article}\\usepackage{amssymb}\\begin{document}\\pagestyle{empty}$\\mathcal {K}\\big (\\mathbf {2}^{X}\\big )\\backslash \\lbrace \\varnothing \\rbrace$\\end{document}<\/jats:styled-content>has a choice set<\/jats:italic>\u201d iff \u201cfor every set<\/jats:italic>X<\/jats:italic>, \\documentclass{article}\\usepackage{amssymb}\\begin{document}\\pagestyle{empty}$\\mathcal {K}\\big (\\mathbf {[0,1]}^{X}\\big )\\backslash \\lbrace \\varnothing \\rbrace$\\end{document}<\/jats:styled-content>has a choice set<\/jats:italic>\u201d iff \u201cfor every product<\/jats:italic>\\documentclass{article}\\usepackage{amssymb}\\begin{document}\\pagestyle{empty}$\\mathbf {X}$\\end{document}<\/jats:styled-content>of finite discrete spaces,<\/jats:italic>\\documentclass{article}\\usepackage{amssymb}\\begin{document}\\pagestyle{empty}$\\mathcal {K}(\\mathbf {X})\\backslash \\lbrace \\varnothing \\rbrace$\\end{document}<\/jats:styled-content> has a choice set<\/jats:italic>\u201d.<\/jats:p>","DOI":"10.1002\/malq.201020039","type":"journal-article","created":{"date-parts":[[2012,4,25]],"date-time":"2012-04-25T12:23:32Z","timestamp":1335356612000},"page":"130-138","source":"Crossref","is-referenced-by-count":4,"title":["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"],"prefix":"10.1002","volume":"58","author":[{"given":"Kyriakos","family":"Keremedis","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2012,4,25]]},"reference":[{"key":"e_1_2_7_2_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02007144"},{"key":"e_1_2_7_3_1","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19850312507"},{"key":"e_1_2_7_4_1","doi-asserted-by":"publisher","DOI":"10.1112\/S0024610798005675"},{"key":"e_1_2_7_5_1","doi-asserted-by":"publisher","DOI":"10.1090\/surv\/059"},{"key":"e_1_2_7_6_1","doi-asserted-by":"publisher","DOI":"10.4064\/ba53-4-1"},{"key":"e_1_2_7_7_1","first-page":"15","article-title":"Extensions of Compactness of Tychonoff Powers of 2 in \\documentclass{article}\\usepackage{amssymb}\\begin{document}\\pagestyle{empty}$\\mathsf {ZF}$\\end{document}","volume":"37","author":"Keremedis K.","year":"2011","journal-title":"Topology 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