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The Abstract Compactness Theorem revisited 1 Abstract. The Abstract Compactness Theorem of Makowsky and Shelah for model the-oretic logics is shown to be an immediate consequence of a general characterization oftopological spaces having [ ; ]-compact products, when applied to spaces of structuresendowed with the natural topology induced by the de…nable classes of a logic L. In thiscontext, the notion of an ultra…lter U being related to L corresponds to U-compactness oftheses spaces. The given characterization of topological productive [ ; ]-compactness mayhave independent interest since it generalizes known results by J. Ginsburg and G. Saks,G. Saks, S. García-Ferreira, and others, for initial - compactness.
Departamento de Matemáticas, Universidad de los Andes 1 Preprint of a paper appeared in pages 131-141of the book: Logic and Foundations of Math- ematics, Selected contributed papers of the Tenth International Congress of Logic, Methodology andPhilosophy of Science, Florence,1995 (Editors: A Cantini, E. Casari, P. Minari). Synthese Library280, Kluwer Academic Publishers 1999.
1991 Mathematics Subject Classi…cation: 03C95, 03C52, 54D30.
It is well known that ×o´s theorem on ultraproducts implies the compactness the- orem of …rst order logic (Frayne, Morel, and Scott 1962 [FMS]). Similarly, utilizingappropriate versions of ×o´s theorem one may prove [ ; ]-compactness of the in…ni-tary logic L Qc+ is the quanti…er “there are at least (2!)+ .”. It is natural to ask then if anyform of compactness in model theoretic logics is associated to some ×o´s-like theo-rem. Makowsky and Shelah 1983 [MSh] have shown the remarkable result that thisis always the case for [ ; ]-compactness of a logic. That is the content of their Abstract Compactness Theorem. A model theoretic logic L is [ ; ]-compact ifand only if there is a ( ; )-regular ultra…lter U over some set I which satis…es thefollowing property: ( ) For any family of structures fAi : i 2 Ig of type A j= [f=U; :::] i¤ fi 2 I : A j= [f(i); :::]g 2 U: (U may be taken always over I = < , or uniform over I = The theorem implies for example that L(Q +1) is [ ; !]-compact if and only (hence, L(Q1) is not [!; !]-compact). Deeper consequences about the compactness spectrum of logics are discussed in Makowsky 1985 [Ma].
The original version in [MSh] is given in terms of extensions of ultrapowers and assumes expansions of vocabularies by binary relation symbols. We have stated thetheorem in terms of ultraproducts because this version implies the original one andholds for a wider family of logics, including monadic logics. For a version in terms ofultralimits see Lipparini 1987 [Li].
An ultra…lter U satisfying the Lo´s-like condition ( ) of the Theorem is said to We show in this paper that the “Abstract Compactness Theorem” and other results on [ ; ]-compactness of logics are purely topological phenomena. They followfrom a characterization of productive [ ; ]-compactness of topological spaces, that is preservation of [ ; ]-compactness by cartesian products, which generalizes analogousresults for productive [ ; !]-compactness by Ginsburg and Saks 1975 [GS], Saks 1978[Sa], and García-Ferreira 1990 [GF]. For this purpose we consider the spaces of …rstorder structures endowed with the topology induced by the “elementary” classes ofa logic L, the key observations being that an ultra…lter U is related to L if andonly these spaces are U-compact in the sense of Saks, and any product of them is[ ; ]-compact when the logic is [ ; ]-compact.
In the topological side, our characterization implies that several properties previ- ously known for logic compactness hold true for productive compactness of topologicalspaces. For example, if is smaller than the …rst measurable cardinal then produc- tive [ ; ]-compactness of a space implies (productive) countable compactness of thespace.
I. [ ; ]-COMPACTNESS AND U-COMPACTNESS OF TOPOLOGICAL SPACES The following natural generalization of the notion of compactness of a topological space was …rst considered by Alexandro¤ and Urysohn in 1929 [AU] and thoroughlystudied later by many people. See the survey papers by Vaughan 1984 [V2] andStephenson 1984 [St], also Nyikos 1992 [N].
De…nition 1.1. A topological space X is [ ; ]-compact, for ! closed subsets of X (of arbitrary power if = 1) such that every sub-family of power less than ( -intersection property), has itself non-empty intersection. [ ; !]-compactness and[1; ]-compactness are usually called initial -compactness and …nal -compactness,respectively.
The reader should be aware that the notation utilized in model theory for [ ; ]- compactness, which we will use in this paper, reverses the notation utilized in thetopological literature.
[1; !]-compactness is (full) compactness, [!; !]-compactness is countable com- pactness, and [1;!1]-compactness is the Lindelöf property. Although [ ; ]- com-pactness does not transfer up or down, for example, ! with the discrete topology istrivially [!1; !1]-compact but not [!; !]-compact and (!1; <) with the order topologyis [!; !]-compact but not [!1; !1]-compact, there are some straightforward transfer LEMMA 1.2. i) X is [ ; ]-compact if and only if it is [ ; ]-compact for any ii) If X is [cof ( ); cof ( )]-compact then it is [ ; ]-compact.
iii) If f : X ! Y is continuous and X is [ ; ]-compact, then f(X) is [ ; ]-compact.
Proof. i) One implication is trivial. For the other, notice that a counterexample minimum is a counterexample to [ ; ]-compactness; (ii) and (iii) follow from the de…nitions.
The product of [ ; ]-compact spaces does not need to be [ ; ]-compact, even for squares. For example, the real line with the topology generated by the intervals[a,b) is a Lindelöf space but its square is not (see Willard 1968 [W]), and the productof two countably compact spaces is not necessarily countably compact (see Vaughan1974 [V1], Hart and Mill 1991 [HM]).
On the positive side, Stephenson and Vaughan 1974 [SV] have shown that [ ; !]- starting with Scaraborough and Stone 1966 [SS] a deep study of spaces with count-ably compact or initial -compact products, and related properties, has been carriedout by Vaughan 1974 [V1], Ginsburg and Saks 1975 [GS], Saks 1978 [Sa], García-Ferreira 1990 [GF], among others. The main tools in this study have been the notionof ultra…lter convergence and compactness, introduced by Berstein 1970 [Be] for ul-tra…lters over !, and extended later by Saks to ultra…lters over uncountable powers.
De…nition 1.3. Let U be an ultra…lter over a set I, then an I-family fai : i 2 Igin a topological space X is said to U-converge to a point x 2 X if and only iffi 2 I : ai 2 V g 2 U for any open neighborhood V of x. We say also that x isan U-limit of fai : i 2 Ig, and write fai : i 2 Ig !U x. A space X will be calledU-compact if and only if any I-family of X has an U-limit in X.
U-limits are not necessarily unique since we do not assume the Hausdor¤ condi- tion. Evidently, fai : i 2 Ig !U x if and only if x is an adherence point in X of theultra…lter a(U) = fS X : fi 2 I : ai 2 Sg 2 Ug in the ordinary sense of topology.
Hence, X is fully compact if and only it is U-compact for any ultra…lter. Contrastingwith [ ; ]-compactness, U-convergence and compactness are preserved by products.
LEMMA 1.4. i) f(ai; ) : i 2 Ig !U (a ) in X is U-compact if and only if each X is U-compact.
In fact, it follows from Saks work for a related compactness property C[ ; ] con- cerning the existence of complete accumulation points (property in general strongerthan [ ; ]-compactness but equivalent to it for ductive [ ; !] -compactness of a space is equivalent to U-compactness with respectto particular families of uniform ultra…lters depending on the space. García-Ferreirahas shown that U-compactness with respect to a single decomposable ultra…lter on is enough. We sumarize this in the next proposition. The last item also follows immediately from Saks work but we have not seen it stated anywhere.
PROPOSITION 1.5 i) (Th. 6.2, Saks 1978 [Sa]; Th. 5.13, Stephenson 1984 [St])All powers of X are [ ; !]-compact if and only if there is a sequence of ultra…ltersfU : ! g; U uniform over , such that X is U -compact for each .
ii) (Prop. 2.15, García-Ferreira 1990 [GF]). All powers of X are [ ; !]-compact if and only if there is a decomposable ultra…lter U over iii) For regular , X has [ ; ]-compact powers if and only if X is U-compact for By Donder 1988 [D], it is consistent that a uniform ultra…lter over decomposable. Therefore, by (ii) and (iii) above, it is consistent that productive [ ; ]-compactness for a regular cardinal is equivalent to productive [ ; !]-compactness.
In order to obtain the Abstract Compactness Theorem (in the next section), we extend the above characterizations to [ ; ]-compactness for arbitrary ; ; utilizing( ; )-regular ultra…lters.
De…nition 1.6 (cf. Keisler 1964 [Ke]). An ultra…lter U over a set I is ( ; )-regularif and only if there is a family F U of power such that \J = Ø for any sub-family J of power : A ( ; !)-regular ultra…lter is usually called -regular in the literature.
It is easy to verify that a uniform ultra…lter over is ( ; ) regular for any regular cardinal Call a class T of spaces productively [ ; ]-compact (in short, p-[ ; ]- compact) if the product of any family of spaces in T is [ ; ]-compact. In particular, a spaceX will be productively [ ; ]-compact if X is [ ; ]-compact for any cardinal . Thenext theorem is our main result.
THEOREM 1.7. The following are equivalent for any class T of topological spaces: i) T is productively [ ; ]-compact.
ii) There exists some ( ; )-regular ultra…lter U such that all the spaces in T are U-compact ( U may be taken always over < , or uniform over The proof is defered to the last section of the paper but we note now the following COROLLARY 1.8. Let T be a class of topological spaces.
is smaller than the …rst measurable cardinal (or arbitrary if no such cardinal exists) and T is p-[ ; ]-compact, then it is p-[!; !]-compact.
ii) If T is p-[ +; +]-compact, then it is p-[ ; ]-compact.
= 1. Then T is p-[ ; ]-compact if and only if it is Proof. i) Let U be the ( ; )-regular ultra…lter over < given by the Theorem such that X is U-compact. By ( ; )-regularity, U is non principal. If U is not(!; !)-regular, then it is !-complete. But the smallest set carrying a !-completenon principal ultra…lter is measurable, and ii) A uniform ultra…lter on + is ( ; )-regular by results of Kanamori 1976 [K] iii) If the compactness condition holds for regular Lemma 1.2(ii). Now apply Lemma 1.2(i).
For the de…nition of model theoretic logic see Lindström 1969 [L] or Ebbinghaus 1985 [E]. The domain of a logic L, Dom(L), is the class of …rst order vocabulariesfor which the class of sentences L( ) is de…ned. Dom(L) will be assumed to allowexpansions of vocabularies by arbitrarily many monadic relation symbols and con-stants, and to be closed under disjoint unions. L( ) will be always a set; that is,we consider small logics only. Appart of Lindström’s axioms ((i)-(v) in [E]), we onlyassume closure under negations, conjunctions, and relativizations. Aj and AjP Awill denote, respectively, the reduct of a structure A to a sub-vocabulary ; and thesubstructure of A induced by the subset P A where P is a monadic relation symbol: 2 Dom(L), a logic L induces a topology in the class E of structures of type ; having for open basis the classes Mod( ); classes are then the L-axiomatizable classes Mod(T ) for some theory T resulting large topological space of structures will be denoted E (L). Although it isproper class, its basis is parametrized by the set L( ) of sentences and so the topologyis also parametrized by a set. Therefore, we may quantify over open classes, classesof open classes, etc., via the parameters, and apply without misgivings most of theordinary concepts and results of topology to these spaces. The spaces E (L) areuniformizable by the canonical uniformity having for basis the classes: where F runs through the …nite theories F L( ), cf. Caicedo 1993 [C1], 1995 [C2].
De…nition 2.1. A logic L is said to be [ ; ]-compact if whenever fT g < is a familyof theories in L( ) such that [ < T The above property is just topological [ ; ]- compactness of the spaces E (L).
The equivalence between this topological notion and the original de…nition of [ ; ]-compactness in Makowsky and Shelah 1983 [Ma-Sh] was …rst noticed by Mannila1983 [M]. It holds for any logic satisfying the closure conditions we have imposed onDom(L).
Given a family of vocabularies f i : i 2 Ig in Dom(L), let i [ fPig where each Pi is a monadic symbol not in Dom(L) is closed under this operation. The function is onto because (Ai)i2I = F ( iAi), where i having for universe the union of the universes of the Ai, with each interpreted by the relations of Ai and Pi interpreted by the universe jAij: Our keyobservation is the following: Hence, L is [ ; ]-compact if and only if the family of topological spaces fE (L) : Dom(L)} is productively [ ; ]-compact.
Proof. Any open subbasic of the product topology in j j j= g = fA : A j= Pj g which is a basic open class due to the reduct axiom and the existence of relativizations in L. By Lemma 1.2 (iii), the [ ; ] compactnessof the space E Now, expressing U-convergence in the spaces E (L) in terms of the basic open classes M od( ) and using that the logic has negations, we have for any ultra…lter Uover I and structures A; Ai (i 2 I) in E (L) (without negations we would have only left to right implication). Therefore, equiva-lence (1) in the Abstract Compactness Theorem applied to sentences yields immedi-ately fAi : i 2 Ig !U A , and so the spaces E (L) must be U-compact if U is relatedto L. In fact, LEMMA 2.3. A ultra…lter U is related to a logic L if and only if the spaces E (L)are U-compact for any Proof. Assume that the spaces E (L) are U-compact. Given a family of structure fAi : i 2 Ig of type , consider the vocabulary each cf is a constant symbol, and de…ne for each …xed j 2 I the following expansionof type where cf is interpreted by f (j). Since E family fAjgj has an U-limit (A ; af ; :::): By (2) this means: A j= [af ; :::] , (A ; af ; :::) j= (cf ; :::) , fj 2 I : Aj j= (cf ; :::)g 2 U , fj 2 I : Aj j= [f(j); :::]g 2 U (x; :::) 2 L( ): Applied to atomic formulae equivalence (3) yields an em- bedding f =U 7! af from iAi=U into A : Therefore, A may be taken to be a trueextension of f =U; and (3) becomes then condition (1), showing Theorem 1.7 applied to the family of spaces T =fE (L) : with lemmas 2.2, 2.3, gives the topological proof of the Abstract CompactnessTheorem.
After Lemma 2.2, any property of productive [ ; ]-compactness of topological spaces may be translated directly to [ ; ]-compactness of logics, without passingby the Abstract Compactness Theorem. For example, the following results fromMakowsky-Shelah 1983 [Ma-Sh] are direct applications of the respective parts ofCorollary 1.8. It follows from this topological proof that they hold for monadiclogics, since Lemma 2.2 only needs expansions of vocabularies by monadic predicatesymbols and relativizations of sentences to monadic predicates. Their original proofsrely instead in the possibility of expanding vocabularies by non monadic relationsymbols.
is below the …rst measurable cardinal then [ ; ]- compactness of a logic implies [!; !]-compactness.
ii) (Th. 3.10 [Ma-Sh]). [ +; +]-compactness of a logic implies [ ; ]-compactness.
iii) (Th. 3.11 [Ma-Sh]). For regular , [ ; ]-compactness of a logic for all regular Remark 2.5. The condition on expansion of vocabularies by constants is not neededin Lemma 2.2, and it may be replaced by closure under the existential quanti…er inLemma 2.3 if the formulae of the logic contain only …nitely many free variables becausethen the constants may be simulated by monadic predicates.
Remark 2.6. To see that the Abstract Compactness Theorem may be stated interms of ultrapowers as in [Ma-Sh] when L allows expansions of vocabularies by binaryrelation symbols and relativization to a variable of a binary relation (for example if Lis closed under substitutions), it is enough to note that the existence of an extensionsatisfying (1) for any U-ultrapower implies the same for any U-ultraproduct. Given afamily fAi : i 2 Ig E , code it in the single structure A = (tiAi; tiQAi; :::; I; R)Q2 containing I as a predicate, and the relation R = [i2Ifig Ai so that Ajfx : R(i; x)g = AI =U satis…es (1), de…ne P = fv 2 B : B j= R[g=U; v]g where g 2 AI is the identity function. Then iAi=U B jP j= [f=U; :::] , B j= [f=U; :::]fv:R(g=U;v)g , fi 2 I : Ai j= [f(i); :::]fv:R(i;v)g 2 U , fi 2 I : Aijfv : R(i; v)g j= [f(i); :::)]g = fi 2 I : Ai j= [f(i); :::]g 2 U: Therefore B jP is the desired extension of iAi=U satisfying (1).
III. CHARACTERIZATION OF PRODUCTIVE [ ; ] -COMPACTNESS In this section we prove Theorem 1.7 in a wide version (Theorem 3.4 below) in- cluding characterizations of productive [ ; ]-compactness in terms of small products,which generalize similar known results for initial -compactness.
LEMMA 3.1. If X is U-compact for a ( ; )-regular ultra…lter U, then X is [ ; ]-compact.
Proof. Let fI g < be a family of elements of U such that the intersection of any of the I ’s is empty. We may assume I = I0. Given a family fF g < of closed sets -intersection property, de…ne Ft = \t2I F for each t 2 I. This set is non-empty, because t belongs to less than at 2 Ft, then J = ft 2 I : at 2 F g 2 U because J hypothesis, fatgt2I U-converges to some x of X; hence, given an open neighborhoodV of x, J = ft 2 I : at 2 V g 2 U. Therefore, ft : at 2 V \ F g = J \ J 2 U for any . Consequently, this set is non-empty, showing that x belongs to the adherence of LEMMA 3.3. i) If X is [ ; ]-compact, then every I-family in X, with I = P ( ; );U-converges for some ( ; )-regular ultra…lter U over I (which depends on the family).
is a regular cardinal and X is [ ; ]-compact then every U-converges for some uniform ultra…lter U over .
Proof. i) Given fat : t 2 Ig in X, let At = fas : t fcl(At) : t2 P ( ; !)g has the -intersection property, because t). Hence, V \ At 6= Ø for any neighborhood V of x and any t 2 P ( ; !).
This implies that a 1(V ) \ [t) = fs 2 I : as 2 V and s fact the family F = fa 1(V ) \ [t) : V open neighborhood of x; t 2 P ( ; !)g has the…nite intersection property. Extend F to an ultra…lter U over I. By construction,fas : s 2 Ig U-converges to x. Moreover, for any ordinal 2 sg = a 1(X) \ [f g) 2 U. But the intersection of -many distinct I ’s is empty many ordinals. This shows that U is ( ; )-regular.
ii) For any -sequence (a ) < in X, let A = fa : and [ ; ]-compactness, there is x 2 \ < cl(A ). Hence, V \ A 6= Ø for any and open neighborhood V of x. By trans…nite induction and regularity of This means that all sets in the family F = fa 1(V ) \ [ ) : x 2 V; < g have power ; hence, F may be extended to an uniform ultra…lter U over , such that (a ) 2 THEOREM 3.4. The following are equivalent for any class T of topological spaces: i) T is productively [ ; ]-compact.
ii) There exists some ( ; )-regular ultra…lter U such that all the spaces in T are U-compact (U may be taken over P ( ; ), or uniform over iii) Any product of 22jP( ; )j many copies of spaces in T is [ ; ]-compact.
) Any product of 22 many copies of spaces in T is [ ; ]- Proof. (ii) ) (i). If each Xr is U-compact then rXr is U-compact by Lemma 1.4(ii), and by ( ; )-regularity of U it follows from Lemma 3.1 that rXr is [ ; ]-compact. This works also if U is uniform over (i) )(iii) ) (ii). Assume that any product of 22jIj many spaces in T is [ ; ]- compact, but there is no ( ; )-regular ultra…lter U over I = P ( ; ) such that all theelements of T are U-compact. Let be the family of all ( ; )-regular ultra…lters over an I-family faU;i : i 2 Ig in some space XU 2 T which does not U-converge. For each i, let i = (aU;i)U 2 U2 XU = X . As at most 22jIj, this space is [ ; ]-compact by hypothesis; then by Lemma 3.3, there isan ultra…lter W 2 such that f i : i 2 Ig W-converges to some By the continuity of the W-projection (Lemma 1.4 (i)), faW;i : i 2 Ig W-convergesto aW in XW, a contradiction.
is regular, then as in the previous proof, working with uniform ultra…lters over I = , utilizing Lemma 3.3 (ii), and recallingthat a uniform ultra…lter on is ( ; )-regular, we get (ii) and a fortiori (i). Now, if ; then the hipotesis implies that the product of 22 spaces in T is [ ; ]-compact, and by the previous observation T is productively[ ; ]-compact. Hence X is p-[ ; ]-compact by Corollary 1.8(iii) which depends onlyon the equivalence (i) , (ii), already proven.
Notice that the equivalence (i),(iv) of the the previous theorem for the cases regular follows already from Th. 2.3 in Saks 1978 [Sa]. Making T = fXg; we obtain generalizations of Th. 5.14 in Stephenson 1984 [St], and Prop.
2.15 in García-Ferreira 1990 [GF]: COROLLARY 3.5. The following are equivalent for any topological space X: ii) X is U-compact for some ultra…lter U (over I = P ( ; ), or uniform over iii) X22jP( ; )j is [ ; ]-compact ( X22 is [ ; ]-compact, if cof( ) iv) XjXjjP( ; )j is [ ; ]-compact ( XjXj is [ ; ]-compact, if cof( ) Proof. Only (iv) ) (i) needs proof. In the proof of Theorem 3.4, (iii) ) (ii), (iv) ) (i), one family fai : i 2 Ig of X may serve as counterexample for the nonconvergence of various ultra…lters U over I; hence, we need to take the factor X inthe power X only once for each possible family: That is, X = XjXjjIj.
Acknowledgements. We thank Paolo Lipparini for making us aware of case (iv)in Theorem 3.4. The main results of this paper were obtained while visiting theUniversidade Estadual de Campinas, Brasil, in 1992, in the course of a seminar ontopological methods in model theory organized with A. M. Sette.
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