Combinatorial and model-theoretical principles related to regularity of ultrafilters and compactness of topological spaces. III

COMBINA TORIAL AND MODEL-THEORETICA L PRINCI P LES RELA TED TO REGULARITY OF UL TRAFIL TERS AND COMP ACTNESS OF TOPOLOGICAL SP ACES. I I I. P AOLO LIPP ARINI Abstract. W e g eneralize the results from [L 2]; in particular the present results apply to singular cardina ls, to o. See [L4, KV, HNV] for definitions and no tation. W e shall need the follow ing theorem f rom [L5]. Theorem 1. If λ is a sing ular c ar dinal, then an ultr afilter is ( λ, λ ) - r e gular if and only if it is either (cf λ, cf λ ) -r e gular or ( λ + , λ + ) -r e gular. Corollary 2. Supp ose that λ is a singular c ar d i n al, an d c onside r the top olo gic al sp ac e X o b tain e d by forming the disjoint union of the top o- lo gic al sp ac es λ + and cf λ , b oth endow e d with the or der top o lo gy. Then, for every ultr afilter D , the sp ac e X is D -c omp act if and only if D is not ( λ, λ ) -r e gular. Thus, X is pr o ductively [ λ ′ , µ ′ ] -c omp act if and only if ther e exists a ( λ ′ , µ ′ ) -r e gular not ( λ, λ ) -r e gular ultr afilter. In p articular, X is not pr o ductively [ λ, λ ] -c o mp act. Pr o of. By Theorem 1, D is not ( λ, λ )-regular if a nd only if it is neither (cf λ, cf λ ) - regular no r ( λ + , λ + )-regular. Hence, b y [L2, Prop o sition 1], and since b oth λ + and cf λ are regular cardinals, D is not ( λ, λ )-regular if and only if b oth λ + and cf λ a r e D -compact. T his is clearly equiv alen t to X b eing D -compact. The last statemen t is immediate from [C2, Theorem 1.7], also stated in [L2, Theorem 2].  2000 Mathematics Subje ct Classific ation. Primary 03E05, 54B10 , 54D20, 54A20; Secondary 0 3E75. Key wor ds and phr ases. Regular ultrafilter s ; compactness of pro ducts o f top o- logical spa ces. The a uthor has rece ived supp o rt fr om MPI a nd GNSA GA. W e wish to ex press our gra titude to X. Ca icedo for stimulating discussions and co rresp ondence. 1 2 COMBINA TORIAL PRINCIPLES, COMP ACTNESS OF SP ACES II I Let 2 = { 0 , 1 } denote the tw o-elemen t s top olog ical space with the discrete top ology . If λ ≤ µ a re cardinals, let 2 µ b e the T ychonoff pro d- uct of µ -man y copies of 2 , and let 2 µ λ denote the subset of 2 µ consisting of a ll those functions h : µ → 2 suc h t ha t |{ α ∈ µ | h ( α ) = 1 }| < λ . In passing, let us men tion that, when µ = ℵ ω , the space 2 µ µ pro vides an example of a linearly Lindel¨ of not Lindel¨ of space. See [AB, Example 4.1]. Compare also [S, Example 4.2 ]. Notice that 2 µ λ is a T yc honoff top ological group with a base of clop en sets. Set theoretically , 2 µ λ is in a one to one corresp ondence (via c haracter- istic functions) with S λ ( µ ), the set of all subse ts of µ of cardinalit y < λ . Since man y prop erties of ultra filters are defined in terms o f S λ ( µ ), for sak e of conv enience, in what follow s w e shall deal with S λ ( µ ), rather than 2 µ λ . Henceforth, we shall deal with the top ology induced on S λ ( µ ) b y the ab ov e corresp ondence. In detail, S λ ( µ ) is endow ed with the smallest top o logy con taining, as op en sets , a ll sets of the form X α = { x ∈ S λ ( µ ) | α ∈ x } ( α v arying in µ ), as w ell as their complemen ts. Th us, a base for the top olo gy consis ts of a ll finite in tersections of the ab ov e sets; that is, the elemen ts of the base are the sets { x ∈ S λ ( µ ) | α 1 ∈ x, α 2 ∈ x, . . . , α n ∈ x, β 1 6∈ x, β 2 6∈ x, . . . , β m 6∈ x } , with n, m v arying in ω and α 1 , . . . , α n , β 1 , . . . , β m v ary- ing µ . Notice that this top ology is finer than the top o logy o n S λ ( µ ) used in [L2]. With the ab ov e top ology , S λ ( µ ) and 2 µ λ are homeomorphic, thu s S λ ( µ ) can b e given the structure of a T yc honoff top olo gical gro up. Notice that if λ ≤ µ then S λ ( µ ) is not [ λ, λ ]-compact. Indeed, for α ∈ µ , let Y α = { x ∈ S λ ( µ ) | α 6∈ x } . If Z ⊆ µ and | Z | = λ then ( Y α ) α ∈ Z is an op en co v er of S λ ( µ ) by λ -many sets, < λ of whic h nev er co v er S λ ( µ ). Prop osition 3. F or ev e ry ultr afilter D and every c ar din al λ , the top o- lo gic al sp a c e S λ ( λ ) is D - c omp act if a n d only if D is not ( λ, λ )-r e gular. Pr o of. Supp ose that D is an ultrafilter ov er I and that S λ ( λ ) is D - compact. F o r ev ery f : I → S λ ( λ ) there exists x ∈ S λ ( λ ) suc h that f ( i ) i ∈ I D -conv erges to x . If α ∈ λ and { i ∈ I | α ∈ f ( i ) } ∈ D then α ∈ x , since otherwise Y = { z ∈ S λ ( λ ) | α 6∈ z } is an op en set con t a ining x , and { i ∈ I | f ( i ) ∈ Y } = { i ∈ I | α 6∈ f ( i ) } 6∈ D , contradicting D - con v ergence. Whence, { α ∈ λ |{ i ∈ I | α ∈ f ( i ) } ∈ D } ⊆ x ∈ S λ ( λ ), and th us x has cardinalit y < λ ; that is, f do es not witness ( λ, λ ) - regularit y of D . Since f has b een chose n arbitrarily , D is not ( λ, λ )-regular. COMBINA TORIAL PRINCIPLES, COMP ACTNESS OF SP ACES II I 3 Con verse ly , supp ose that D o v er I is not ( λ, λ )- regular, and let f : I → S λ ( λ ). Then x = { α ∈ λ |{ i ∈ I | α ∈ f ( i ) } ∈ D } has cardinality < λ and hence is in S λ ( λ ). W e show that f D -conv erges to x . Indeed, let Y b e a neighbor ho o d of x : we hav e to show that { i ∈ I | f ( i ) ∈ Y } ∈ D . Without loss of generalit y , w e can supp ose that Y is an elemen t of the base of S λ ( λ ), that is, Y has the form { z ∈ S λ ( λ ) | α 1 ∈ z , α 2 ∈ z , . . . , α n ∈ z , β 1 6∈ z , β 2 6∈ z , . . . , β m 6∈ z } . Since D is closed under finite in tersections, then { i ∈ I | f ( i ) ∈ Y } ∈ D if and only if { i ∈ I | α 1 ∈ f ( i ) } ∈ D and { i ∈ I | α 2 ∈ f ( i ) } ∈ D and. . . and { i ∈ I | α n ∈ f ( i ) } ∈ D a nd { i ∈ I | β 1 6∈ f ( i ) } ∈ D and. . . and { i ∈ I | β m 6∈ f ( i ) } ∈ D . But all the ab ov e sets are actualy in D , b y the definition o f x and since x ∈ Y and D is an ultra filter; thus f D -conv erges to x . Since f w as arbitrary , ev ery f : I → S λ ( λ ) D -conv erges, a nd th us S λ ( λ ) is D - compact.  Corollary 4. The sp ac e S λ ( λ ) is pr o ductively [ λ ′ , µ ′ ] -c omp act if and only if ther e e x i sts a ( λ ′ , µ ′ ) -r e gular no t- ( λ, λ ) -r e gular ultr afilter. Pr o of. Immediate from Prop osition 3 and [C2, Theorem 1.7 ].  In the statemen ts o f the next theorems the w o rd “pro ductiv ely”, when included within paren theses, can b e equiv alen tly inserted or omit- ted. Theorem 5. F or al l infinite c ar dinals λ , µ , κ , the fo l lowing ar e e quiv- alent: (i) Every pr o ductively [ λ, µ ] -c omp act top olo gic al sp ac e is (pr o duc- tively) [ κ, κ ] -c omp ac t. (ii) Every pr o ductively [ λ, µ ] -c omp act family of top olo gic al sp ac es is pr o ductively [ κ, κ ] -c om p act. (iii) Every ( λ, µ ) -r e gular ultr afil ter is ( κ, κ ) -r e gular. (iv) Every pr o ductively [ λ, µ ] -c omp act Hausdorff no rm al top olo gic al sp ac e with a b ase of clop en se ts is pr o ductively [ κ, κ ] -c o m p act. (v) Every p r o ductively [ λ, µ ] -c omp act T ycho n off top olo g i c al gr o up with a b ase of clop en sets is (pr o ductively) [ κ, κ ] -c omp act. If κ is r e gular, then the pr e c e ding c onditions ar e also e quivalent to: (vi) Every pr o ductively [ λ, µ ] -c omp act Hausdorff no rm al top olo gic al sp ac e with a b ase of clop en se ts is [ κ, κ ] -c o m p act. Pr o of. Let us denote by (i) p Condition (i) when the second o ccurrence of the w ord “pro ductiv ely” is included, and simply b y ( i) when it is omitted. Similarly , for conditio n (v). The equiv alence of (i)-(iii) has been pro v ed in [L2, Theorem 1], where it has also b een prov ed that, for κ regular, they ar e equiv a len t to (vi). 4 COMBINA TORIAL PRINCIPLES, COMP ACTNESS OF SP ACES II I Since (ii) ⇒ (i) p ⇒ (i) are t rivial, w e get t ha t (i), (ii), (iii) , (i) p are all equiv alen t, a nd equiv alen t to (vi) for κ regular. (ii) ⇒ (iv) and (ii) ⇒ (v) p ⇒ (v) are trivial. If (iii) fa ils, then there is a ( λ, µ )-regular ultr a filter whic h is not ( κ, κ )-regular, th us, for κ singular, the space X of Corollary 2 is pro- ductiv ely [ λ, µ ]- compact. F or κ regular, tak e X = κ with the order top ology (see [L2]). X is Hausdorff, no rmal, with a ba se of clop en sets, but not pro ductiv ely [ κ, κ ]-compact, again by Coro llary 2, th us (iv) fails. W e hav e pro v ed (iv) ⇒ (iii). (v) ⇒ (iii) is similar, using Corollary 4, since S κ ( κ ) is not [ κ, κ ]- compact.  Theorem 6. F or al l infinite c ar dinals λ , µ , and for an y family ( κ i ) i ∈ I of infinite c ar dinals, the fol lowing ar e e q uiva l e nt: (i) Every pr o ductively [ λ, µ ] -c omp act top olo gic al sp ac e is (pr o duc- tively) [ κ i , κ i ] -c omp act for some i ∈ I . (ii) Every pr o ductively [ λ, µ ] -c omp act family of top olo gic al sp ac es is pr o ductively [ κ i , κ i ] -c omp act for some i ∈ I . (iii) Every ( λ, µ ) -r e gular ultr afilter is ( κ i , κ i ) -r e gular fo r some i ∈ I . (iv) Every pr o ductively [ λ, µ ] -c omp act Hausdorff no rm al top olo gic al sp ac e with a b a s e of clop en sets i s pr o ductively [ κ i , κ i ] -c omp act for so me i ∈ I . (v) Every p r o ductively [ λ, µ ] -c omp act T ycho n off top olo g i c al gr o up with a b ase of clop en sets is (pr o ductively) [ κ i , κ i ] -c omp act for some i ∈ I . If every κ i is r e gular, then the pr e c e ding c ond i tion s ar e al s o e quivale nt to: (vi) Every pr o ductively [ λ, µ ] -c omp act Hausdorff no rm al top olo gic al sp ac e with a b ase of clop en se ts is [ κ i , κ i ] -c omp act for some i ∈ I . Pr o of. The equiv alence of (i)-(iii) has b een prov ed in [L2, Theorem 3], th us, arguing as in the pro of of Theorem 5, w e get that (i), (ii), (iii), (i) p are all equiv alent. (ii) ⇒ (iv) ⇒ (vi) and (ii) ⇒ (v) p ⇒ (v) are trivial. If (iii) fails, then there is a ( λ, µ )-regular ultrafilter D whic h for no i ∈ I is ( κ i , κ i )-regular. By Prop osition 3 , for eve ry i ∈ I the top ological space S κ i ( κ i ) is D -compact. Hence X = Q i ∈ I S κ i ( κ i ) is D -compact, th us pro ductiv ely [ λ, µ ]- compact, by [C2, Theorem 1.7 ]. Ho w ev er, X is a T yc honoff top o logical gro up with a base of clop en sets whic h for no i ∈ I is [ κ i , κ i ]-compact, th us (v) fails. W e hav e pro v ed (v) ⇒ (iii). The pro ofs of (iv) ⇒ ( iii) and (vi) ⇒ (iii) are similar, using the next prop osition. If (iii) fails, then there is a ( λ, µ )-regular ultrafilter D whic h for no i ∈ I is ( κ i , κ i )-regular. By the pro of of Theorem 5, for COMBINA TORIAL PRINCIPLES, COMP ACTNESS OF SP ACES II I 5 ev ery i ∈ I w e hav e a D -compact top ological space X i whic h falsify 5(iv), resp., 5(vi). Then the space X = { x } ˙ ∪ ˙ S i ∈ I X i w e shall construct in the next definition is D -compact, thus pro ductiv ely [ λ, µ ]- compact, b y [C2, Theorem 1.7], and mak es ( iv), resp., (vi), fail.  Definition 7. Giv en a family ( X i ) i ∈ I of top ological spaces, construct their F r e chet disjo i n t union X = { x } ˙ ∪ ˙ S i ∈ I X i as follo ws. Set theoretically , X is the unio n of (disjoint copies) of the X i ’s, plus a new elemen t x whic h b elong s to no X i . The top olo g y on X is the smallest top ology whic h contains each op en set of eac h X i , and whic h con tains { x } ˙ ∪ ˙ S i ∈ E X i , for ev ery E ⊆ I such that I \ E is finite. Prop osition 8. I f ( X i ) i ∈ I is a family o f top o lo gic al sp a c es, then their F r e ch e t disjoi n t union X = { x } ˙ ∪ ˙ S i ∈ I X i is T 0 , T 1 , Hausdorff, r e gular, normal, D -c om p act (for a given ultr afilter D ), [ λ, µ ] -c omp act (for given infinite c ar dinals λ and µ ), has a b ase of cl o p en sets if and only i f so is (has) e ach X i . Pr o of. Straightforw ard. W e shall comment only o n regularit y , normal- it y and D -compactness. F or regularity and normalit y , just observ e that if C is closed in X and C has nonempt y in tersection with infinitely many X i ’s, t hen x ∈ C . As for D -compactness, suppo se D is o v er J and that eac h X i is D - compact. Let ( y j ) j ∈ J b e a sequence of elemen ts of X . If { j ∈ J | y j ∈ S i ∈ F X i } 6∈ D holds for ev ery F ⊆ I , then ( y j ) j ∈ J D -conv erges to x . Otherwise, sinc e D is a n ultrafilter, hence ω -complete, there exists some i ∈ I suc h that { j ∈ J | y j ∈ X i } ∈ D . But then ( y j ) j ∈ J D -conv erges to some p oin t o f X i , since X i is supp o sed to be D -compact.  When κ is singular of cofinality ω , Condition (vi) in Theorem 5 is equiv alent to the other conditions. When eac h κ i is either a regular cardinal, or a singular cardinal of cofinalit y ω , then Condition (vi) in Theorem 6 . is equiv alen t to the other conditions. Pro ofs shall b e given elsewhere . Reference s [AB] A. V. Arha ng el’skii, R. Z. Buzyako v a, On line arly Lindel¨ of and st ro ngly dis- cr etely Lindel¨ of sp ac es. P ro c. Amer. Math. So c. 12 7 (1999), no. 8 , 244 9–245 8. [C1] X. Caicedo, On pr o ductive [ κ, λ ] -c omp actness, or the Abstr act Comp actn ess The or em r evisite d , manuscript (1995 ). [C2] X. 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Lippar ini, Comp act factors in fin al ly c omp act pr o ducts of top olo gic al sp ac es , T op olog y and its Applications, 153 (2006 ), 1365–1 382. [L4] P . Lipparini, Combinatorial and mo del-the or etic al principles r elate d to r e g- ularity of ultra filters and c omp actness of top olo gic al s p ac es. I , arXiv:080 3.3498 ; II. :0804.1 4 45 (2008). [L5] P . Lippa rini, De c omp osable ultr afilters and p ossible c ofinalities , in print on Notre Dame Jo urnal of F orma l Logic. [S] R. M. Stephenso n, Initial ly κ -c omp act and r elate d sp ac es , Chapter 13 in [KV], 603–6 32. Dip ar timento di Ma tema tica, Viale della Rice rca Scientifica, II Universit accia Romana ccia (Tor Verga t a), I-00133 ROME I T AL Y URL : h ttp:// www.ma t.uniroma2.it/~lipparin