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On Reflection of Stationary Sets

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On Reflection of Stationary Sets F U N D AM E NTA nnoindentMATHEMATICAEF U N D AM E NTA 1 40 open parenthesis 1 9 9 2 closing parenthesis nnoindentOn reflectionMATHEMATICAE of stationary sets by n centerlineQi .. F e nf g1 .. 40 open ( parenthesis 1 9 9 2 ) Singaporeg closing parenthesis .. and .. Menachem .. M a g i d o r .. open parenthesis JerusalemF U N closing D AM E parenthesis NTA n centerlineAbstract periodfOnMATHEMATICAE .. reflection We show that of there stationary are stationary sets subsetsg of uncountable spaces which do not reflect period 1 40 ( 1 9 9 2 ) n centerlineS 0 period ..fby Thereg are many usefulOn and interestingreflection stationary of stationary reflection setsprinci hyphen ples formulated and studied in the current research concerningby new existence n centerlineaxioms of setfQiQi theorynquad F and e nF combinatorial g e n( Singapore g nquad aspects )( Singapore ofand infinityMenachem period ) nquad Mand a gn iquad d o rMenachem( Jerusalemnquad ) M a g i d o r nquad ( Jerusalem ) g Notably comma oneAbstract of the first . reflectionWe show principles that there studied are stationary i s the following subsets of : uncountable spaces which Abstractopen parenthesis . nquaddo A notWe closing reflect show parenthesis . that there .... If kappa are stationary greater equal aleph subsets sub 2 of .... uncountable is regular comma spaces S subset which equaldo not kappa reflect is stationaryx .0 . commaThere .... then are theremany is useful some and alpha interesting less kappa stationary reflection princi - ples such that S capformulated alpha is stationary and studied in alpha in the period current research concerning new existence nS Then0 . manynquad stationaryThereaxioms ofare reflection set many theory principles useful and combinatorial of and the following interesting aspects form of have infinity stationary . reflection princi − plesbeen formulated formulated and and studiedNotably studied period , one inof the the first current reflection research principles studied concerning i s the following new existence : open parenthesis(A) B closingIf κ parenthesis≥ @2 is .... regular If kappa;S ⊆ greaterκ is stationary equal aleph , sub then 2comma there is S some subsetα < equal κ open squarennoindent bracketaxioms kappa closing of set square theory bracket andsuch to the combinatorialthat powerS \ α ofis aleph stationary subaspects 0 ....in isα: of stationary infinity comma . .... then there is X in open square bracketThen many kappa stationary closing square reflection bracket principles to the power of the of following aleph sub form 1 .... have such been that formu- n centerlineS cap openf squareNotablylated bracket and , studied one X closing of . the square first bracket reflection to the power principles of aleph sub 0 studied .. is stationary i s the in the following space : g @0 @1 open square bracket(B) X closingIf κ square≥ @2;S bracket⊆ [κ] to theis stationarypower of aleph , subthen 0 there period is X 2 [κ] such that nnoindentApparently(A) comman ..h f open i l l parenthesisI f $ Sn\kappa[X B]@ closing0 isngeq stationaryparenthesisnaleph in i sthe much spacef 2 moreg[X$ ] powerful@0n: h f i l la principleis regular than open $ , parenthesisS nsubseteq A closing parenthesisnkappaApparently$ is , isperiod ( stationary B ..) i A s natmuch hyphen more , n h powerful f i l l then a principle there than is ( some A ) is . $ n Aalpha nat - < nkappaural question$ wouldural question b e : .... canwould there b e be : some useful can generalizations there be some useful of open generalizations parenthesis B ofclosing ( B ) parenthesis t o t o uncountable spaces ? n centerlineuncountablef spacessuch ? that $We S showncap in thisn notealpha that$ one is must stationary b e very careful in in $ formulatingnalpha such. $ g We show in thisa notething that . one must b e very careful in formulating such Thena thing many period stationaryOur reflection set - theoretic principles usage i s standard of the . followingAll undefined form t erminology have i s taken beenOur set formulated hyphenfrom theoretic and [ 4 ] . studiedusage i s standard . period .. All undefined t erminology i s taken from open square bracket 4 closingx1 . squareLet bracketκ > period ! b e a regular cardinal . For any cardinal nnoindentS 1 period(B) .. Letλ ..n≥h kappa f iκ, l l greaterIlet f $ omegankappa .. b en ageq regularnaleph cardinal periodf 2 g .. For,S any ..n cardinalsubseteq .. lambda[ nkappa ] ^f naleph f 0 gg$ n h f i l l is stationary , n h f i l l then there is $X n in greater equal kappaPκ comma(λ) = .. letfx ⊆ λ j j x j < κg: By a nontrivial filter on Pκ(λ) we mean [ Pn subkappa kappa open]a ^f parenthesis naleph lambdaf 1 gg closing$ n h parenthesis f i l l such = open that brace x subset equal lambda bar bar x bar less kappa closingcollection brace periodF of .... subsets By a of ....Pκ nontrivial(λ) satisfying filter the on following P sub kappa conditions open: parenthesis lambda closing parenthesisn centerline ....f we$ mean S ancap [ X ] ^f naleph f 0 gg$ nquad is stationary in the space $ [ X ] ^f naleph f 0 gg . $ g collection F of subsets of(1) P8 subx 2 kappaPκ(λ)( openPκ(λ) parenthesis− fxg) 2 F lambda; ?element closing− parenthesisnegationslashF; satisfying the following conditions : ( 2 ) if A; B 2 F and A \ B ⊆ C; then C 2 F: n hspaceopen parenthesis∗fn f i l l g 1Apparently closing parenthesis , nquad forall( x in B P ) sub i kappa s much open more parenthesis powerful lambda a closing principle parenthesis than open ( A ) is . nquad A nat − parenthesis P sub kappa open parenthesis lambda closing parenthesis minus open brace x closing brace closing parenthesisnnoindent inural F semicolon question varnothing would element-negationslash b e : n h f i l l can F comma there be some useful generalizations of ( B ) t o open parenthesis 2 closing parenthesis if A comma B in F and A cap B subset equal C comma then C in F periodnnoindent uncountable spaces $ ? $ n hspace ∗fn f i l l gWe show in this note that one must b e very careful in formulating such nnoindent a thing . n hspace ∗fn f i l l gOur s e t − theoretic usage i s standard . nquad All undefined t erminology i s taken nnoindent from [ 4 ] . n hspace ∗fn f i l l gnS 1 . nquad Let nquad $ nkappa > nomega $ nquad b e a regular cardinal . nquad For any nquad c a r d i n a l nquad $ nlambda ngeq nkappa , $ nquad l e t nnoindent $ P f nkappa g ( nlambda ) = nf x nsubseteq nlambda nmid nmid x nmid < nkappa ng . $ n h f i l l By a n h f i l l nontrivial filter on $ P f nkappa g ( nlambda ) $ n h f i l l we mean a nnoindent collection $ F $ of subsets of $ P f nkappa g ( nlambda ) $ satisfying the following conditions : n [ ( 1 ) n f o r a l l x n in P f nkappa g ( nlambda )(P f nkappa g ( nlambda ) − nf x ng ) n in F; n varnothing element−negationslash F, n ] n centerline f( 2 ) i f $ A , B n in F $ and $ A ncap B nsubseteq C , $ then $ C n in F . $ g 1 76 .. Q period F e n g and M period .. Magi d o r nnoindentUnless otherwise1 76 n spquad ecifiedQ.FengandM. comma all filters are assumednquad t oMagi be nontrivial d o r period A filter F on P sub kappa open parenthesis lambda closing parenthesis i s kappa hyphen complete if F i s closedn centerline under intersectionsf Unless otherwiseof sp ecified , all filters are assumed t o be nontrivial . g less than kappa many of it s members period F is normal if F i s closed under diagonal n hspaceintersections∗fn f i comma l l gAfilter i period e period $F$ comma on whenever $P f angbracketleft nkappa g ( A subnlambda alpha bar alpha) $ less i lambda s $ rightnkappa −angbracket$ complete i s a sequence1 if 76 $ Qfrom F . F $ F e n comma i g ands closed M then . Magi under d o r intersections of vartriangle A sub alphaUnless = open otherwise brace x in sp P ecified sub kappa , all filters open parenthesis are assumed lambda t o be closing nontrivial parenthesis . bar forall nnoindent l e s s than $ nkappa $ many of it s members $ . F$ is normal if $F$ alpha in x x in A sub alpha closingA filter braceF inon FPκ period(λ) i s κ− complete if F i s closed under intersections of i sIf closedF is a filter underless on P than diagonal subκ kappamany open of it s parenthesis members :F lambdais normal closingif parenthesisF i s closed comma under we diagonal denote by F to the power of plus all Fintersections hyphen positive , i . sete . ,s whenever comma i periodhAα j α e < period λi i s comma a sequence from F; then nnoindentF to the powerintersections of plus = open , brace i . A e subset . , wheneverequal P sub kappa $ n langle open parenthesisA f lambda nalpha closingg nmid parenthesisnalpha < nlambda nrangle $ i s a sequence from $F , $ then bar forall C in F A cap C negationslash-equalM Aα varnothing= fx 2 Pκ( closingλ) j 8α brace2 xx 2 periodAαg 2 F: Then a filter F is normal if and only if for each A in F to+ the power of plus and for every choice If F is a filter on Pκ(λ); we denote by F all F − positive set s , i . e . , n [ functionn vartriangle f on A open A parenthesisf nalpha i periodg = e periodnf commax n fin open parenthesisP f nkappa x closingg ( parenthesisnlambda in x for) nmid n f o r a l l nalpha n in x x n in A f nalpha g ng n in F.
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