; TeX output 2013.01.06:1441 y ? I!Dt G G cmr17A7tnoteonTqychono8's7tTheoremandAxiomof -8Choice A"XQ cmr12Ko jiNuida 2&Published:8Janruary22,2012ν HLastUpSdate:8Janruary6,2013!ٍ !K9t: cmbx9Abstractd6o cmr9TheL aimKofthisnoteistogiv9esomeobserv|rationonastandardproAof W toTdeduceAxiomofChoicefromT9ychono q'sTTheorem.6M K`y cmr10Incthiscnote,webasicallydealwiththeaxiomsZFO! cmsy7 *ofsettheory*,which > meansUUtheZermelo{F*raenkelsettheoryZFN:excepttheAxiomofFoundation؍ d4 !", cmsy108 b> cmmi10x(9y[ٲ(y"2x)!9y(y"2x8^9zp(z72x^z2y[ٲ))) :M In\ethis\fnote,^)wesay\fthataclassK_ofsetsisB ': cmti10Bdownwar}'dgcloseddif,^)forany\eset> A2KUTand0[any0ZsetB˲forwhich0ZthereexistsaninjectivemapBG,UX!A,7itfollows> thatBl2K$.Intuitively*,thismeansthatKisaclassofcardinalnumbGerswith> thelpropGertylthatjX jjY8j2KimpliesljX j2K$.F*orlexample,r~thelclassesCm#R cmss10CSet> ofallsets,CFinite>ofallnitesets,andCCountable15)ofallcountablesets(i.e.,sets> AUUforwhichjAj@ٓR cmr70|s)UUaredownwardclosedclassesofsets.M Onytheotherhand,wesaythataclassT)$oftopGologicalspacesisaBtop}'ological> pr}'opertyif,for .anyX2TڲandanytopGological /spaceYFwhich /ishomeomorphic> tosnX ,itfollowsthatY2T.&{Namely*,weidentifyatopGologicalsopropertysn(inusual> sense)withtheclassofalltopGologicalspaceshavingthepropGerty*.N8Forexample,> theclassesCT*opofalltopGologicalspaces,CT1iofallT1|s-spaces,andCHausdor.ofall> Hausdorspacesare topGologicalproperties.1Amemberof atopologicalproperty> T isUUsaidtobGeaTB-sp}'ace.M In+whatfollows,4>we+assume+thatKPisadownward+closedclassofsetsandT> isUUatopGologicalproperty*.qWeUUdenethefollowingpropGositions:z> ACLq(K$)`wLetAbGeafamilyofnon-emptysetswithA<2K$.EThenthereexistsaW choiceifunctionforjA,i.e.,aimapf:A! u cmex10S Asatisfyingthatf(A)2AW forUUeveryA2A.μ> ACEqX(K$)mThesameasAC (K$),exceptthatallmembGersofAaresuppGosedtoW haveUUequalcardinality*.ν> AMCZLetAbGeafamilyofnon-emptysets.5Thenthereexistsa\multiplechoiceW function"forA,1i.e.,amapf:AA!A2:# cmex7S۟^AsatisfyingthatforeachA2AA,W f(A)UUisanitenon-emptysubsetofA. 1 *y ? > AMCEqfThesameasAMCj,YexceptthatallmembGersofAaresuppGosedtohave W equalUUcardinality*.YI> T(T;K$)f2MLet钸A鑲bGeafamilyofcompactT-spaces..}ThenanyopGencover鑸WofW theUUproGducttopologicalspaceQqƸAhasasubcoverUUW }^0withW }^0gθ2K$.> THomeoc(T;K$) ynThesameasT(T;K$),AexceptthatallmembGersofAaresup-W pGosedUUtobehomeomorphictoeachother.> F*or~.example,5AC(CSet s)is~-theAxiomofChoice(AC),AC(CCountable*#)~.istheAxiom> of5CountableChoice(ACC),6AMC?istheAxiomofMultiple6Choice(AMC),and> T(CT*op#;CFinite\q)ɟistheɞTychono 'sɟTheorem.C5NotethatAC;g(CFiniteɲ)isatheoremof> ZFJ T(soRisQACEq