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

COMBINA TORIAL AND MODEL-THEORETICA L PRINCI PLES RELA TED TO REGULARI TY OF UL TRAFIL TERS AND COM P ACTNESS OF TOPOLOGICAL SP A CES. II. P AOLO LIPP ARINI Abstract. W e find ma n y conditio ns equiv alen t to the model- theoretical prop erty λ κ ⇒ µ in tro duced in [L1]. Our conditions in- volv e uniformity of ultrafilters, compactness prop erties of pro ducts of top ological spaces a nd the existence of cer ta in infinite matric e s . See P art I [L7] or [CN, CK, KM, KV, HNV] for unexplained notation. According to [L1], if λ ≥ µ are infinite regular cardinals, and κ is a cardinal, λ κ ⇒ µ means that the mo del h λ, <, γ i γ <λ has an expansion A in a language with at most κ new sym b ols such that whenev er B ≡ A and B has a n elemen t x suc h that B | = γ < x for ev ery γ < λ , then B has an elemen t y suc h that B | = α < y < µ for ev ery α < µ . An ultrafilter D ov er λ is said to b e uniform if and only if ev ery mem b er of D has cardinalit y λ . If λ is a regular cardinal, then it is ob vious that an ultrafilter D is uniform o v er λ if and only if the in terv al [0 , γ ] 6∈ D , for ev ery γ < λ , if and only if the interv al ( γ , λ ) is in D , for ev ery γ < λ . Th us, if D is an ultrafilter o ver some regular cardinal λ , and if I d D denotes the D -class of the iden tit y function on λ , then D is uniform o v er λ if and only if in the mo del C = Q D A w e hav e that d ( γ ) < I d D for ev ery γ < λ . Here, d denotes the elemen t ary em b edding. If D is a n ultrafilter o ver I , and f : I → J , then f ( D ) is the ultrafilter o v er J defined by: Y ∈ f ( D ) if and only if f − 1 ( Y ) ∈ D . If κ, λ are infinite cardinals, a top ological space is said to b e [ κ, λ ] - c omp act if and only if ev ery op en cov er b y at most λ sets has a s ub co v er 2000 Mathematics Subje ct Classific ation. P rimary 03 C20, 0 3E05, 54B10, 54D20; Secondary 0 3C55, 03 C98. Key wor ds and phr ases. Elementary extensions of car dinals with order; infinite matrices; uniform, regular , decomp osable ultra filters; compactness of pro ducts o f top ological s pa ces. The author ha s received s uppo r t fro m MPI and GNSA GA. W e wish to e xpressed our gra titude to X. Caicedo for stimulating discussio ns and corresp ondence. 1 2 COMBINA TORIAL PRINCIPLES, COMP ACTNESS OF S P ACES b y less than κ sets. No separatio n axiom is needed to pro v e the results of the presen t pap er. Theorem 1. Supp ose that λ ≥ µ ar e infi nite r e gular c ar dinals, and κ ≥ λ is an infinite c ar dinal. Then the fol lowing c onditions ar e e quivalent. (a) λ κ ⇒ µ h o lds. (b) T her e a r e κ f unction s ( f β ) β <κ fr om λ to µ such that whenever D is an ultr afilter uniform over λ then ther e exis ts some β < κ such that f β ( D ) is uniform over µ . (b ′ ) Ther e ar e κ functions ( f β ) β <κ fr om λ to µ for which the fol lo w ing holds: fo r every function g : κ → µ ther e exists some finite set F ⊆ κ such that    T β ∈ F f − 1 β ([0 , g ( β )))    < λ . (c) Ther e is a family ( B α,β ) α<µ,β <κ of subsets of λ such that: (i) F or every β < κ , S α<µ B α,β = λ ; (ii) F o r every β < κ and α ≤ α ′ < µ , B α,β ⊆ B α ′ ,β ; (iii) F or every function g : κ → µ ther e exists a finite subse t F ⊆ κ such that | T β ∈ F B g ( β ) ,β | < λ . (d) Whenever ( X β ) β <κ is a family of top o lo gic al s p ac es such that no X β is [ µ, µ ] -c omp act, then X = Q β <κ X β is not [ λ, λ ] -c omp act. (e) The top olo gic a l s p ac e µ κ is not [ λ, λ ] -c omp act, wher e µ is endowe d with the top olo gy whos e op en sets ar e the intervals [0 , α ) ( α ≤ µ ), and µ κ is endowe d with the T ychon off top olo gy. R emark 2 . An ana logue of T heorem 1 holds for the more general notion ( λ, µ ) κ ⇒ ( λ ′ , µ ′ ) in tro duced in [L2] (see also [L3, Section 0 ]). Details shall b e presen ted elsewhere. F o r this more general notion, t he equiv- alence of conditions analog ue to (a) and (b) ab ov e has b een stated in [L5]. There w e also stated the analogue of (b) ⇒ (d). Pr o of. (a) ⇒ (b). Let A b e an expansion of h λ, <, γ i γ <λ witnessing λ κ ⇒ µ . Without loss of generalit y w e can assume that A has Sk o lem func- tions (see [CK, Section 3.3]). Indeed, since κ ≥ λ , adding Sk olem functions to A inv olv es adding at most κ new sym b ols. Consider the set of all functions f : λ → µ which are definable in A . En umerate them as ( f β ) β <κ . W e are going to sho w that these functions witness (b). Indeed, let D b e an ultrafilter uniform ov er λ . Consider the D -class I d D of the iden tit y function on λ . Since D is uniform o v er λ , in the mo del C = Q D A we hav e that d ( γ ) < I d D for eve ry γ < λ , where d denotes the elemen tary em b edding. Let B b e the Sk olem h ull of I d D COMBINA TORIAL PRINCIPLES, COMP ACTNESS OF S P ACES 3 in C . By Lo ˇ s Theorem, C ≡ A . Since A has Skolem functions, B ≡ C [CK, Prop osition 3.3 .2]. By transitivity , B ≡ A . Since A witnesse s λ κ ⇒ µ , then B has an elemen t y D suc h that B | = α < y D < µ fo r ev ery α < µ . Since B is the Sk olem h ull of I d D in C , w e hav e y D = f ( I d D ), that is, y D = f D , for some function f : λ → λ definable in A . Since f is definable, then a lso the followin g function f ′ is definable: f ′ ( γ ) = ( f ( γ ) if f ( γ ) < µ 0 if f ( γ ) ≥ µ Since B | = y D < µ , then { γ < λ | y ( γ ) < µ } ∈ D . Since y D = f D , { γ < λ | y ( γ ) = f ( γ ) } ∈ D . Hence, { γ < λ | y ( γ ) = f ′ ( γ ) } ∈ D , b eing larger than the in tersection of tw o sets in D . Th us, y D = f ′ D . Since f ′ : λ → µ and f ′ is definable in A , then f = f β for some β < κ , th us y D = ( f β ) D . W e need to show that D ′ = f β ( D ) is uniform o v er µ . Indeed, for ev ery α 0 < µ , a nd since B | = α 0 < y D , then { γ < λ | α 0 < y ( γ ) } ∈ D ; that is, { γ < λ | α 0 < f β ( γ ) } ∈ D , that is, { α < µ | α 0 < α } ∈ D ′ , and this implies that D ′ is uniform ov er µ , since µ is regular. (b) ⇒ (a ). Suppo se w e hav e functions ( f β ) β <κ as giv en by (b). Expand h λ, <, γ i γ <λ to a mo del A b y adding, for eac h β < κ , a new function sym b o l represen ting f β (b y abuse of notation, in what follows w e shall write f β b oth for the f unction itself a nd f or the sym b ol that represen ts it). Supp ose tha t B ≡ A and B has an elemen t x suc h that B | = γ < x for ev ery γ < λ . F or ev ery form ula φ ( z ) with just one v ariable z in the languag e of A let E φ = { γ < λ | A | = φ ( γ ) } . Let F = { E φ | B | = φ ( x ) } . Since the in tersection of an y t wo mem b ers of F is still in F , and ∅ 6∈ F , then F can b e extended to an ultra filter D on λ . F or eve ry γ 0 < λ , consider the for m ula φ ( z ) ≡ γ 0 < z . W e get E φ = { γ < λ | A | = γ 0 < γ } = ( γ 0 , λ ). On the other side, since B | = γ 0 < x , then by the definition of F we hav e E φ = ( γ 0 , λ ) ∈ F ⊆ D . Th us, D is uniform o ver λ . By (b), f β ( D ) is uniform ov er µ , for some β < κ . This means that ( α 0 , µ ) ∈ f β ( D ), for ev ery α 0 < µ . That is, { γ < λ | α 0 < f β ( γ ) } ∈ D for ev ery α 0 < µ . F or ev ery α 0 < µ , consider the form ula ψ ( z ) ≡ α 0 < f β ( z ). By the previous paragraph, E ψ ∈ D . Notice that E ¬ ψ is the complemen t of E ψ in λ . Since D is prop er, and E ψ ∈ D , then E ¬ ψ 6∈ D . Since D extends 4 COMBINA TORIAL PRINCIPLES, COMP ACTNESS OF S P ACES F , and either E ψ ∈ F or E ¬ ψ ∈ F , w e necessarily ha v e E ψ ∈ F , that is, B | = ψ ( x ), that is, B | = α 0 < f β ( x ). Since α 0 < µ has b een chosen arbitrarily , w e hav e that B | = α 0 < f β ( x ) f or ev ery α 0 < µ . Moreov er, since f β : λ → µ , and B ≡ A , then B | = f β ( x ) < µ . Th us, we ha v e pro v ed that B has an elem en t y = f β ( x ) suc h that B | = α < y < µ for ev ery α < µ . (b) ⇔ (b ′ ) follows fro m Lemma 3 b elo w. (b ′ ) ⇒ ( c). Supp o se that w e hav e functions ( f β ) β <κ as giv en by (b ′ ). F or α < µ and β < κ , define B α,β = f − 1 β ([0 , α )). The family ( B α,β ) α<µ,β <κ trivially satisfies Conditions (i) a nd (ii) . Moreo ver, Condition (iii) is clearly equiv alen t to the condition imp osed on the f β ’s in (b ′ ). (c) ⇒ (b ′ ). Supp ose w e are giv en the family ( B α,β ) α<µ,β <κ from (c). F or β < κ and γ < λ , define f β ( γ ) to be the smallest ordinal α < µ suc h that γ ∈ B α,β (suc h an α exists b ecause o f (i)). Because of Condition (ii), w e hav e that B α,β = f − 1 β ([0 , α ]), for α < µ and β < κ . Th us Condition (iii) implies that for ev ery function g : κ → µ t here exists s ome finite set F ⊆ κ suc h that    T β ∈ F f − 1 β ([0 , g ( β )])    < λ . A fortiori,    T β ∈ F f − 1 β ([0 , g ( β )))    < λ , th us (b ′ ) holds. The equiv alence of Conditions (c)-(e) has b een pro v ed in P art I [L7, Theorem 2].  Lemma 3. S upp ose that λ ≥ µ ar e infinite r e gular c ar dinals, and κ is a c ar dinal. Supp os e that ( f β ) β <κ is a given set of functions fr om λ to µ . Then the fol lowing ar e e quivalent. (a) Whenever D is an ultr afilter uniform over λ then ther e exists some β < κ such that f β ( D ) is uniform over µ . (b) F or eve ry function g : κ → µ ther e exists som e finite set F ⊆ κ such that    T β ∈ F f − 1 β ([0 , g ( β )))    < λ . Pr o of. W e sho w that the negation of (a) is equiv a len t to the negation of (b). Indeed, (a) is false if and only if there exists an ultrafilter D uniform o v er λ such that for ev ery β < κ f β ( D ) is not uniform ov er µ . This means that f or eve ry β < κ there exists some g ( β ) < µ such that [ g ( β ) , µ ) 6∈ f β ( D ), t hat is, f − 1 β ([ g ( β ) , µ )) 6∈ D , that is, f − 1 β ([0 , g ( β ))) ∈ D . Th us, there exists some D whic h mak es (a) false if and only if there exists some function g : κ → µ suc h that the set { f − 1 β ([0 , g ( β ))) | β < κ } ∪ { [ γ , λ ) | γ < λ } has the finite interse ction prop ert y . Equiv alently , COMBINA TORIAL PRINCIPLES, COMP ACTNESS OF S P ACES 5 there ex ists some function g : κ → µ suc h tha t fo r eve ry F ⊆ κ the cardinalit y of T β ∈ F f − 1 β ([0 , g ( β ))) is equal to λ (since λ is regular). This is exactly the negation of (b).  Reference s [C1] X. Ca icedo, On pr o ductive [ κ, λ ] -c omp actness, or the Abstr act Comp actness The or em r evisite d , manuscript (1995). [C2] X. 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