On generalized topological spaces

On generalized top ological spaces Ar tur Pięk osz Abstract In this pap er a systemati c study of the catego ry GTS of gener- alized top olo gical spaces (in the sense of H. Delfs and M. Knebusch) and their strictly co n tin uous mappings b egins. Some completeness and cocompleteness resu lts are ac hiev e d. Generalized top ological spaces h elp to reconstruct the imp ortan t elemen ts of the theory of lo c ally definable and w eakly definable space s in the wide co n text of w eakly top ological s tru ct ures. 2000 MS Classific ation: 54A05, 18F10, 03C65. Key wor ds and fr ases: generalize d top ologica l space, Grothendiec k top olog y , top olo gical structure. 1 In t r o duction Generalized to p ology in the sense o d H. Delfs and M. Knebusc h is an un- kno wn c hapter of general top ology . In fact, it is a generalization of the classical concept of top ology . The aim o f this pap er is to start the system - atic s tudy of generalized top ology in this sense . (Do not mix with other meanings of ”generalized top ology” appearing in the literature.) This concept w as define d in [3] and helped t o dev elop a semialgebraic v er- sion of homotop y theory . In 1991, in his pap er [8], M. Knebusc h suggested that his theory of lo cally semialgebraic spaces (dev elop ed in [3 ] together with H. Delfs) and w eakly semialgebraic spaces (dev elop ed in [7]) could b e generalized to the o-minimal conte xt. (Here lo cally definable and w eakly definable spaces allo w to mak e constructions analogical to th at kno wn from the traditional homotop y theory .) Success ful generalization of the theory of Delfs and Knebusc h to the case of o- min imal expansions of fields in [11] op ens a question if similar homotopy theories can b e dev eloped b y the use of generalized top ology and other ideas from [3, 7]. Moreo v er, ev en if in many cases a full-fledged homotopy theory will not b e av ailable, the use of lo cally definable and w eakly definable spaces o v er v arious structures will b e imp ortan t. And ev en on the pure top ological lev el, generalized top ology is an intere sting notion, w o rth to b e studied. Notice that the concept of generalized top ology se ems not to b e understoo d, since the pap er [1] suggests its non-existence (see [11] for a discuss ion) despite a heavy use of the theory of lo cally semialgebraic s paces from [3]. The T -spaces of [4] ha v e natural generalized top ologies, and it is con vieni en t to see them as generalized top ological spaces. The category GTS of generalized top ological spaces and their strictly con tin uous mappings may b e seen as an alternativ e to the usually used cat- egory T op of top ological spaces and con tin uous mappings. It or ig inates from the categorical concept of Grothendiec k top ology , and con tains T op 1 as a full sub category . In general, the third o rder concept o d a generalized top ological space is m uc h more difficult to study than the second order con- cept of a top ological space. Only the use of GTS allo w ed to glue infinitely man y definable sets (considered with the ir natural o-minimal top ologies) to pro duce lo cally defin able spaces in [3] and [11] and w eakly definabl e spaces in [7] and [11]. Man y of the pro ofs of [3] and [7] are purely top ological (naturally in the sense of the generalized top ology), and it is imp ortan t to extract the information on par t icular lev els of structure. The theory of in- finite gluings of definable sets can be reconstructed to a large exten t in the setting of w eakly top ological structures. This pap er is a contin uation of the pap er [11], whic h w as dev oted to extending homotopy theory to the case of spaces ov er o-minimal expan- sions of fields. T he presen t pap er is m uc h more general and giv es a nice axiomatization a nd a basic theory of the category GTS . The new (rela- tiv e to traditional top ology) concept of admissibilit y is exp lained, and the pap er deals with main (generalized) top ological concepts as: s mall sets, bases, connectedness , completeness, paracompactness, Lindel¨ ofness, sepa- ration axioms, and v a r io us concept o f discreteness. W e pro v e completeness and co completeness of the full subcategory Small of sm all spac es, and fi- nite completeness of the sub categories LSS , WSS 1 . Then GTS is used to build some natural categories of spaces o v er mo del-theoretic structures. A natural setting here for to pological considerations is assuming a top ology on the underlying set M o f a mo del-theoretic structure M , and insisting on regarding the pro duct top ologies on the cartesian p o w ers of M . Suc h struc- tures are called in this pap er w eakly top ological. They are more g eneral than so called first order top ological structures of [12, 10] and structures with a de finable topolo g y of [13]. W e pro v e co complete ness of the category Space ( M ) of space s o v er a w eakly topo lo g ic al M , a nd finite completene ss of its full subcategories ADS ( M ) , DS ( M ) , LDS ( M ) , WDS 1 ( M ) . The author hop es that fro m no w on generalized top ology (hidden in the language of lo cally semialgebraic spaces of [3], and weak ly semialgebraic spaces of [7]) will b e dev elop ed without constraints . Some op en questions are suggested to the reader. Notation. F or families o f sets U , V , we will use usual op erations ∪ , ∩ , \ on sets, and analogical op erations o n families of sets, for ex ample U ∩ V = { U ∩ V : U ∈ U , V ∈ V } , U ×V = { U × V : U ∈ U , V ∈ V } . In particular, w e will denote V ∩ U = { V }∩ U for a set V . 2 Grothend i e c k t op ology Here w e remind what a Gro thendiec k top ology is. The r eader ma y consult b oo k s lik e [2, 6, 9]. Let C b e a small category . Consider the category of preshea v es o f sets on C , denoted by P sh ( C ) or ˆ C , whic h is the category of contra v ariant functors from C to Sets . Let us remind the fundamen tal fact: 2 F act 2.1 ( Y oneda lemma, weak version ) . F unctor C ∋ C 7→ H om ( − , C ) ∈ ˆ C is ful l and faithful, so we c an c onsider C ⊆ ˆ C . T o define a Grothendiec k top ology the follo wing notion is usually used: a siev e S o n an ob ject C is a sub ob ject of H om ( − , C ) as an ob ject of ˆ C . Since a siev e on C is a presheaf of sets of morphisms with common co domain C , this ma y b e translated (with a small abus e of language) in to: S is a set of morphisms with co domain C suc h that f ∈ S implies f ◦ g ∈ S , if only f ◦ g is meaningful. The largest siev e o n C is H om ( − , C ) , the smallest is ∅ . A Grothendieck topology J on C is a function C 7→ J ( C ) , with J ( C ) a set of siev es on C suc h that the following axioms ho ld: (iden tit y/nonemp t yness) for eac h C , H om ( − , C ) ∈ J ( C ) ; (stabilit y/base c hange) if S ∈ J ( C ) and f : D → C is a morphism of C , then f ∗ S = { g | f ◦ g ∈ S } ∈ J ( D ) ; (transitivit y/lo cal c haracter) if S ∈ J ( C ) and R is a siev e on C suc h that f ∗ R ∈ J ( D ) for eac h f ∈ S , f : D → C , then R ∈ J ( C ) . Elemen ts of J ( C ) are called cov ering siev es . P air ( C , J ) is called a Grothendiec k site . The ab o v e axioms imply the conditions (se e section I II.2 of [9]): (saturation) if S ∈ J ( C ) and R is a siev e containin g S , then R ∈ J ( C ) ; (in tersection) if R, S ∈ J ( C ) , then R ∩ S ∈ J ( C ) . It follow s tha t each J ( C ) is a filter (not necessary prop er) on the latt ice S u b ˆ C H o m ( − , C ) of siev es on C . If the category C has pul lb acks , then instead of cov ering siev es, w e can sp eak ab out co v ering families o f morphisms (g e nerating resp e ctiv e siev es), so the axioms ma y b e reform ulated as: (iden tit y/isomorph ism) for each C , { id C } is a cov ering family (stated also as: for eac h isomorphis m f : D → C , { f } is a co ve ring family); (base chang e) if { f i : U i → U } i is a co v ering family , and g : W → U an y morphism, then { π 2 i : U i × U W → W } i is a co verin g fa mi ly; (lo cal c haracter) if { f i : U i → U } i is a cov ering family , and { g ij : V ij → U i } j are co v ering families, then { f i ◦ g ij : V ij → U } ij is a co v ering family; and usually the follo wing is added (see Definition 16.1.2 of [6]): (saturation) if { f i : U i → U } i is a co v ering family , and each of the f i ’s factorizes through an elemen t of { g j : V j → U } j , then { g j : V j → U } j is a co v ering family . Alternativ ely (see section 6 .7 of [2]) authors consider saturated and non- saturated Grothendiec k top ologies, but for a G rothend iec k site saturation is usually assumed. Grothendiec k top ology allo ws to define shea v es (of sets). A sheaf is suc h a presheaf F that for each co v ering family { U i → U } i in the resp ectiv e diagram F ( U ) e − → Y i F ( U i ) p 1 − → − → p 2 Y i,j F ( U i × U U j ) the induced morphism e is the equalizer of the standardly considered pair of morphisms p 1 , p 2 (cf. section I I I.4 in [9]). 3 A Grothendiec k top ology is su b canoni cal if ev ery represen ta ble presheaf is a s heaf, so in this case w e ma y , b y ide n tifying the ob jects of C with their respectiv e represen table preshea v es, consider C ⊆ S h ( C ) ⊆ P sh ( C ) = ˆ C . Grothendiec k top ologies used in practice are usually subcanonical. 3 Generalized top ological space s This section is dev oted to in tro ducing a nice axiomatization and basic prop- erties of generalized top ological sp aces. F or any set X , w e hav e a b o olean algebra P ( X ) of subsets o f X , so this algebra of sets may be treated as a sm all category with inclus ions as morphisms. In this category fib ered pro ducts are the same as (binary) pro ducts and the same as (binary) in tersec tions, so P ( X ) has pullbac ks. W e wan t to in tro duce a f ul l subcategory O p of P ( X ) , consisting of “op en subsets ” of X . The n we w ant to introduce a sub canonical Grothendiec k top ology on this category . Subcanonicalit y means in this setting that for eac h co v ering family of morphisms (whic h ma y b e identi fied with a family of subsets of a giv en set, since morphisms are inclusions ) the ob ject cov ered b y the family is the suprem um of this family in the smaller category O p . This leads to the notion of a generalized top ological space in tro duce d b y H. Delfs and M. K nebusc h in [3 ]: a generalized top ological space ( gts ) is a s et X together with a family of subsets ◦ T ( X ) of X , called op en sets , and a family of op en families Cov X , called admissible families o r admissib le (op en) co v erings , s uc h that: (A1) ∅ , X ∈ ◦ T ( X ) (the empt y set and the whole space are are op en), (A2) if U 1 , U 2 ∈ ◦ T ( X ) then U 1 ∪ U 2 , U 1 ∩ U 2 ∈ ◦ T ( X ) (finite unions and finite in tersections of op en sets are o pen), (A3) if { U i } i ∈ I ⊂ ◦ T ( X ) and I is finite, t hen { U i } i ∈ I ∈ Cov X (finite families of op en se ts are admissible), The ab o v e three axioms are a strengthening of the iden tity axiom. They also insure that the smaller category has pullback s. These thre e axioms ma y b e colle ctiv ely called the finiteness axiom. (A4) if { U i } i ∈ I ∈ Co v X then S i ∈ I U i ∈ ◦ T ( X ) (the union of an admissible family is op en), This axiom ma y be called co-sub canonicalit y . T ogether with sub- canonicalit y , it means that admissible families a re co v erings (in the traditional sense) of their unions, whic h is imp osed b y the notation of [3]: { U i } i ∈ I ∈ Cov X ( U ) iff U is the union of { U i } i ∈ I . Sub canonic alit y and co-sub canonicalit y ma y b e collectiv ely called t he naturalit y axiom. 4 (A5) if { U i } i ∈ I ∈ Co v X , V ⊂ S i ∈ I U i , and V ∈ ◦ T ( X ) , then { V ∩ U i } i ∈ I ∈ Co v X (the in tersections of an a dmissible family with an o pen subset of the union of the family form an admiss ible family), This is the stabilit y axiom. (A6) if { U i } i ∈ I ∈ Co v X and for eac h i ∈ I there is { V ij } j ∈ J i ∈ Co v X suc h that S j ∈ J i V ij = U i , then { V ij } i ∈ I j ∈ J i ∈ Co v X (all mem b ers of admissible co v erings of mem b ers of an admissible family form together an admissible family), This is the transitivit y axiom. (A7) if { U i } i ∈ I ⊂ ◦ T ( X ) , { V j } j ∈ J ∈ Co v X , S j ∈ J V j = S i ∈ I U i , and ∀ j ∈ J ∃ i ∈ I : V j ⊂ U i , then { U i } i ∈ I ∈ Cov X (a coarsen ing, with the same union, of an admissible family is admissible ), This is the saturation axiom. (A8) if { U i } i ∈ I ∈ Co v X , V ⊂ S i ∈ I U i and V ∩ U i ∈ ◦ T ( X ) for each i , then V ∈ ◦ T ( X ) (if a subset of the union of an admissible family fo r ms op en in tersec tions with the mem b ers o f the family then the subset is op en). This axiom ma y be called regularit y . Both saturation and regularit y ha v e a smo othing c haracter. Satura- tion may b e ac hiev ed by adding coarsenings of admissible co v erings for any gts . Regularity ma y b e achi ev ed for the class of lo cally small spaces (see b elo w for the definition) b y adding “lo cally op en” subsets and allo wing locally essen tially finite co ve rings (see section I.1 in [3]). The ab o v e axioms ma y b e restated shortly in the follo wing w a y . A generalized top ological space is a triple ( M , O p M , C ov M ) , where M is an y set, O p M ⊆ P ( M ) , and C o v M ⊆ P ( O p M ) , suc h that the followin g axioms are satisfied: ( finiteness ) if U ∈ F in ( O p M ) , then S U , T U ∈ O p M , U ∈ C ov M , ( stabilit y ) if V ∈ O p M , U ∈ C ov M , then V ∩ U ∈ C ov M , ( transitivit y ) if Φ ∈ P ( C ov M ) , S Φ ∈ C ov M , then S Φ ∈ C ov M , ( saturation ) if U ∈ C ov M , V ∈ P ( O p M ) , U  V , then V ∈ C ov M , ( regularit y ) if W ∈ P ( M ) , U ∈ C ov M , W ∩ U ∈ P ( O p M ) , then W ∩ ( S U ) ∈ O p M . In the ab o v e F in ( · ) is the family of finite subsets of a giv en set, and U  V means: these tw o families hav e the same union a nd U is a refinemen t of V (so V is a coarsening of U ). Notice that O p M = S C ov M = S C ov M , hence one can define a generalized top ological space as just a pair ( M , C ov M ) . The naturality axiom do es not app ear, since o ur in terpretation of the families O p M and C o v M is in tended to giv e a G r o the ndiec k site. (Eac h mem b er of C ov M co v ers its union.) A strictly con tinu ous mapping betw een gts es is suc h a mapping that the preimage o f an admissible family is an admissible family . This, in particular, means that the preimage o f an op en set should b e op en. (Strictly 5 con tin uous mappings may b e view ed as morphisms of sites, compare section 6.7 of [2] and section 1 7.2 of [6].) The gts es tog e ther with the strictly con tin uous mappings f o rm a category called here GTS . Isomorphisms of GTS will b e calle d strict homeomorphis ms . W e will say that a family U is essen tially finite (coun table) if some finite (countable ) subfamily U 0 ⊆ U co ve rs the union of U (i. e. S U 0 = S U ). Example 3.1. We hav e the fol lowing si mple examples of g t s es: 1. The sp ac e R n alg , wher e the close d sets ar e the a lgebr a ic subsets of R n , and the admissib l e c overings ar e the essential ly finite op en famili e s (this is R n with the Zariski top olo gy). 2. The sp ac e C n alg , wher e the close d sets ar e the a lgebr a ic subsets of C n , and the admissib l e c overings ar e the essential ly finite op en famili e s (this is C n with the Zariski top olo gy). 3. The sp ac e R n salg , wher e the op en sets ar e the op en semi a lgebr a i c sub- sets of R n , and the adm issible c overin g s ar e the essential ly fi nite op en families. 4. The sp ac e R n san , wher e the op en sets ar e the o p en semianalytic subsets of R n , and the admissible c overings ar e the op en families essential ly finite on b ounde d sets of R n . 5. The sp ac e R n suban , wher e t he op en sets ar e the op en sub an alyt ic subsets of R n , and the admissible c overings ar e the op en families essential ly finite on b ounde d sets of R n . 6. The sp a c e R n top , the usual top olo gic al sp ac e R n . 7. The sp ac e R n ts , wher e the op en sets ar e the sets op e n in the usual top olo gy, and the admissib l e c overings ar e the essential ly finite op en families. 8. F or e ach top olo gic al sp ac e ( X, τ ) , we c an take O p X = τ , and as C ov X the essential ly c ountable op en famil i e s . The func tion sin : R → R is an endomorphism of R top and of R san , but not an endomorphi s m of R salg . Understanding o f the new concept o f admissibilit y is giv en in the fol- lo wing tw o prop ositions and t he nex t remark. First of all, notice that, for eac h op en f a m ily U , saturatio n implies: U ∈ C ov M iff U ∪ { ∅ } ∈ C ov M iff U \ { ∅ } ∈ C o v M . Prop osition 3.2. I f U and V ar e admissib le, then: a) U ∪ V is admi s s ible, b) U ∪ V is admissible, 6 c) U ∩V is admissi b l e . Mor e over, if S U , S V ar e op en , ( S U ) ∩ ( S V ) = ∅ and U ∪ V is admissible, then d) U and V ar e admissible . Pr o of. a) Let Φ = {U , V } . Then, b y finiteness, one has S Φ = { S U , S V } ∈ F in ( O p M ) ⊆ C ov M . By transitivit y , w e get S Φ = U ∪ V ∈ C ov M . b) Notice that U ∪ V  U ∪ V and use saturation. c) Consider Φ = {U ∩ V | V ∈ V } . Since S Φ = { ( S U ) ∩ V | V ∈ V } = ( S U ) ∩ V ∈ C ov M , w e ha ve S Φ = U ∩ V ∈ C ov M . d) This follo ws fro m stability a nd saturation, since ( S U ) ∩ ( U ∪ V )  U , and similarly for V . Prop osition 3.3 (omitting o f admissible unions) . Assume the op en famili e s U , V j ( j ∈ J ) ar e admissible and the fam ily U ∪ S j ( V j ∪ { S V j } ) is admiss ible. Then U ∪ S j V j is adm issible. Pr o of. Notice that U ∪ S j ( V j ∪ { S V j } ) is a refinemen t o f U ∪ S j { S V j } , so the second family is admissible . By applying transitivit y to the family of families {{ U } : U ∈ U } ∪ {V j : j ∈ J } , w e get admiss ibilit y of the family U ∪ S j V j . Remark 3.4. Notic e that a subfam ily of an admisssible family may no t b e a d m issible, even if they have the same unio n . Similarly if U and V ar e admissible, then U ∩ V may not b e admiss i b le. F or ex a mple, c onsid er R to b e an affine semia l g e br aic sp ac e (se e [11]). T ake U = { ( 1 n , 1 − 1 n ) : n ≥ 3 } , V = { (0 , 1) } , W = { (0 , 2) } . Then U ∪ V , U ∪ W ∈ C ov R , but U = ( U ∪ V ) ∩ ( U ∪ W ) / ∈ C ov R . On the other ha n d, an op en sup e rfa mily (with the same union) of an admissible family is always admissible by satur ation. W e call a subset K of a gts X small if for eac h admissible co v ering U of an y op en U , the set K ∩ U is co ve red b y finitely man y mem b ers of U . (W e sa y in this case that the co verin g U is essen tially finite on K or on K ∩ U .) The class of small spaces forms a full subcategory Small of GTS . Prop osition 3.5. 1 ) A s ubse t of a smal l set is smal l. 2) The image of a smal l set by a strictly c ontinuous mappin g is smal l. Pr o of. 1) T ak e any admissible (op en) cov ering U of U . Assume L ⊆ K and K is small. Since U is essen tially finite on K ∩ U , it is also e ssen tially finite on its subset L ∩ U . 2) Ass ume f : X → Y is strictly con tin uous, and K ⊆ X is small. T ak e an admissible op en co v ering V of V in Y . Then f − 1 ( V ) is essen tially finite on K , and f ( K ) ∩ V = f ( K ∩ f − 1 ( V )) is essen tially finite. Prop osition 3.6. In any gts if an op en family is essential ly finite (on its union), then this family is admissible. 7 Pr o of. First notice that if an op en family U has the largest elemen t U , then { U } and U are refinemen ts of eac h other. Th us, b y saturation, U is admissible. No w if an o pen fa m ily U has a finite subcov er U 0 (of its union), then for eac h V ∈ U 0 the family V ∩ U has the largest elemen t V , and is admissible. Since U 0 is admissible , also U 0 ∩ U is admissible b y t r a ns itivit y , and U is admissible b y saturation. Corollary 3.7. An op en family of a sm al l sp ac e is admissible if and only if it is essential ly finite. Example 3.8. Each top o l o gic al sp ac e may b e c on sider e d as a g ts . A d- missible c overing s ar e understo o d as any c overings in the tr aditional sense. Each c o ntinuous function is her e strictly c ontinuous. Thus T op is a ful l sub c ate gory of GTS . Let us denote the small subsets of X b y S m X . W e alwa ys hav e F in ( X ) ⊆ S m X ⊆ P ( X ) . In the case o f X small, w e get S m X = P ( X ) . Denote S mO p X = S m X ∩ O p X . Prop osition 3.9. The usual euclide an sp ac e w ith the usual top olo gy (d e - note d R n top ) is a gts in which al l smal l subsets ar e finite. I n p articular, the c omp act interval [0 , 1] i n R top is not sm al l. Pr o of. A small subset K of the (top ological) space R n top is quasi-compact and all of its subsets are also quasi-compact. But R n top is Hausdorff. Since all subsets of K are compact, K is a discrete set, and finally a finite set. Example 3.10 (non-examples) . Given a top olo g ic al sp ac e, we c ould c o n - sider on ly singletons of op en se t s as admissible c ov e rings of these sets. This would giv e so-c al le d indiscr ete Gr othendie ck top olo gy , but not (in gen - er al) a gts , b e c a u se of (A3) . On the other hand, if we c onsider e d al l fa m - ilies of op en subsets as c overings o f an op en set, the r es ulting discr ete Gr othendie ck top olo gy would not (in g ener a l ) b e sub c anonic al, henc e not a g t s . Let us remind t wo in terestin g examples of gts es. Example 3.11 (“the subanalytic site,” Remark 23 in [11]) . If M is a r e al analytic manifold, then we c an set: a) an op en subs et o f a gts me ans an o p en sub a nalytic subset; b) an admissible cov ering of a gts me ans a family that is ess e n t ial ly finite on c om p ac t subsets. Example 3.12 (another “site,” Remark 23 in [11]) . In the situation as in Example 3.11, set: a) an op en subs et o f a gts me ans any sub analytic subset; b) an admissible cov ering of a gts me ans a family that is ess e n t ial ly finite on c om p ac t subsets. 8 Notice that compact sets are small in Examples 3.11 and 3 .12 , but not necess arily small in Example 3.8 (b ecaus e an op en subse t of a compact set is not usually compact). F or an y gts X and a small or o pen Y ⊆ X we can induce a gts on Y (then Y will b e calle d a subspace of X ) in the follo wing w ay : a) the open subsets of the gts Y are exactly the traces o f op en subsets of the gts X on Y , b) the admissible families o f the gts Y are exactly the traces of a dmissible families of the gts X on Y . This works for op en subsets , since op ennes s and a dmissibilit y in an op en subspace is equiv alent to op ennes s and admissibilit y in the whole space. It w orks for small subsets b ec ause in a small space “admissible” means exactly “essen tially finite”. In general, the problem with transitivit y , saturation and regularit y arises. A subse t Y of a gts X is closed if its complemen t Y c is op en, and is lo cally closed if it is an in tersection of a closed set and a n open set. A subset Y of a gts X is constructible if it is a Bo olean com bination of op en sets. Remind that eac h constructible set is a finite union of lo cally closed sets. If { ( O p α , C ov α ) } α ∈ A is a family of generalized top ologies on a set X , then their inte section ( T α O p α , T α C ov α ) is a generalized top ology . Th us, for any family { V i } i ∈ I of subsets of a gts X and an y fa m ily {U j } j ∈ J of subfamilies of { V i } i ∈ I (this is not restrictiv e b ec ause we ma y enlagre this family if needed), w e can sp eak ab out the generalized top o logy generated b y { V i } i ∈ I and {U j } j ∈ J . A basis of the generalized top ology is suc h a family of op en sets that an y op en set is an admissible union of elemen ts from the basis. (The notion of the basis of a top ology is a special case of the notion of the basis of the generalized top ology .) Notice that if B is a basis of a gts ( X , O p, C ov ) , then ( B , C ov B ) generates ( O p, C ov ) , where C ov B = C ov ∩ P ( B ) . A subset O ⊆ X will b e called w eakly op en if it is a union of op en subsets of the gts X , and weakly closed if its complemen t is w eakly op en. The w eakly op en s ubsets form a top ology gene rated by t he op en subsets. The follo wing example show s that a con tin uous mapping ( in the gen- erated topo logy) that maps small sets onto small sets may not b e strictly con tin uous. Example 3.13. T ake the r e al line R and define the op en sets a s the finite unions of op en intervals with endp oints b eing r ation a l numb ers or infinities. Define the a dmissible c overings as the essential ly finite c overings. We get a smal l gts . The mappin g R ∋ x 7→ r x ∈ R , fo r r / ∈ Q , is c ontinuous but not strictly c ontinuous. A subset Y ⊆ X will b e called dense in X if its w eak closure (i.e. closure in the generated top ology) is equal to X . A gts X is separable if there is a coun t a ble dens e subset of X . W e can in tro duc e the coun tabilit y axioms: a gts X satisfies the first axiom of countab ilit y if eac h p oin t of X has a coun table basis of open 9 neigh borho o ds (suc h a family ma y easily be made admissible), and satisfies the second axiom of coun tabilit y if there is a coun table basis of the generalized top ology . Example 3.14. The semia l g e br aic r e a l line R salg has a c ountable b asis of the gener ate d top olo gy { ( p, q ) : p < q , p, q ∈ Q } wh ich i s not a b asis of the gener alize d top olo gy. Of course, eac h g t s satisfying the second axiom of coun tabilit y is sepa- rable. The separation axioms in GTS ha v e w eak and strong v ersions. A gts X will b e called: a) wea kly T 1 if for eac h x ∈ X and y ∈ X \ { x } there is an op en set U suc h that x ∈ U and y / ∈ U ; b) st r on gly T 1 if eac h si ngleton is a closed subset of X ; c) w eakly Hausdorff if for each pair of differen t p oin ts x, y ∈ X there are op en disjoin t U, V suc h that x ∈ U and y ∈ V ; d) strongly Hausdorff if it is we akly Hausdorff and strongly T 1 ; e) w eakly regular if for each p oin t x ∈ X and eac h subset F of X not con taining x and b eing either c losed or a singleton there are open di sjoin t sets U, V suc h that x ∈ U and F ⊆ V ; f ) strongly regular if it is strongly T 1 and w eakly regular; g) w eakly normal if for t w o disjoint sets F , G b ein g eac h either closed o r a singleton there a re op en dis join t sets U, V suc h that F ⊆ U and G ⊆ V ; h) strongly normal if it is strongly T 1 and w eakly normal. It is clear that a gts is w eakly T 1 (w eakly Hausdorff ) if and only if the generated top ology of X is T 1 (Hausdorff, resp ectiv ely). Eac h w eakly regular gts has a regular generated topo logy . Example 3.15 (strange small sp ace) . L et X b e an infinite gts wher e o p en sets ar e exactly fi nite sets o r the whole sp ac e, and adm i s sible c overings ar e exactly essential ly finite o p en families. Then X is not str on g ly T 1 , but the gener ate d top o l o gy is discr ete. Prop osition 3.16. Th e c ate gory Small is c omplete. Pr o of. The pro duct of a set of small spaces { X α } α ∈ A is the cartesian pro d- uct of sets X = Q α ∈ A X α with the g e neralized top ology describ ed as follow s: op en sets are finite unions o f full cylinders with base s being finite cartesian pro ducts of op en sets and the admis sible co verings a re the essen tially finite co v erings. Then obv iously the pro jections π α : X → X α are s trictly con tin- uous, and for a sys tem of strictly con tin uous mappings f α : Y → X α , the induced mapping ( f α ) α : Y → X is strictly con tinu ous. The ex istence of equalizers for pairs of parallel strictly con tinuous func - tions b et we en small spaces is obvious , since each subset of a small space forms a subspace. There exist t wo inclusion functors i s : Small → GTS and i t : T op → GTS , whic h are full and fa ithful. It is easy to chec k that the functor i s admits a left adjoin t sm , whic h will b e called the smallification functor: sm ( X ) = X sm , sm ( f ) = f , 10 where O p X sm = O p X , C ov X sm = E ssF in ( O p X ) . Here E ssF in ( · ) denotes the family of essen tially finite families. Similarly , the functor i t admits a righ t a djo in t top , w hic h w ill b e called the top ologization functor: top ( X ) = X top , top ( f ) = f , where O p X top = τ ( O p X ) , C ov X top = P ( τ ( O p X )) . Here τ ( · ) denotes the generated top ology . Hence sm preserv es colimits and top preserv es limits. Example 3.17 (top ologization of the b oolean algebra of definable sets) . L et M b e a first or der mathem atic al structur e (i. e. a set M with som e distinguishe d r e l a t ions, c onstants and functions). Then for e ach A ⊆ M and n ∈ N , the family D ef n ( M , A ) of A -definable subsets of M n forms a b o ole an algebr a. Con sider O p = D ef n ( M , A ) , and C ov = E ssF in ( O p ) . Then the gener ate d top olo gy τ ( O p ) is the family of W -definable ov e r A subsets of M n , and the c omp l e m ents of memb ers of τ ( O p ) (i. e. the we akly close d sets) ar e the typ e definable over A subsets of M n . (In p r actic e, in mo del the ory often b ounds ar e s e t on the c ar dinality of the family of op en sets forming a W -definable set.) F act 3.18 (I.2 (p. 11) in [3]) . The c ate gory GTS is c o c omplete. Corollary 3.19. Small is a c o c omplete c ate gory. An op en map ping is a strictly con tin uous mapping b et w een gts es suc h that the image of any o pen subset of the domain is an op en subset of the range. It is clear that the canonical pro jections from the pro duct of small spaces to its factors are op en mappings. Similarly , a closed mapping maps closed subsets onto closed subsets. The canonical pro jections from the pro duct of small spaces to its factors are closed mappings. Each strict homeomorphism is b oth an op en mapping and a closed mapping. A lo cal strict homeomorphi sm is a stric tly con tin uous mapping f : X → Y for whic h there is an o pen admissible cov ering X = S α ∈ A X α suc h that eac h f | X α is an o pen mapping and a strict homeomorphism on to its image. The next example sho ws that a lo cal strict homeomorphism may not b e an op en mapping. Example 3.20. A lo c al ly semialge br aic c overing p : R loc → S 1 (in the sen se of Example 13 in [11]) is a lo c al strict home omorphism but it is neither an op en m a p ping nor a close d mapping. After p assin g to the str ong top olo gies, we get the mapping p top : ( R loc ) top → ( S 1 ) top , which is op en but n ot close d. F or an y family { X α } α ∈ A of gts es suc h that eac h in tersection X α ∩ X β is an open subs pace both in X α and in X β , the re is a uniq ue gts X ha ving all X α s as op en subspaces and the family { X α } α ∈ A admissible. This is clear for finite families, and follo ws from the F act 3.18 for an y families, since then X is the colim it of t he sys tem of finite unions of spaces X α partially ordered b y inclusion. In particular, if an admissible cov ering U of X is giv en, then the space X is uniquely determined as the admissible union of the family 11 U of op en sub spaces. In suc h a situation, w e will write X = a S U . If the mem b ers of U = { U i } i ∈ I are pairwise disjoin t, then the resultin g space X is caled the direct ( generalized t opological) sum (or copro duct) of the elemen ts of U , and will b e denoted X = L i ∈ I U i . Notice that the n eac h U i is a ls o closed. Moreo v er, eac h family of unions o f U i s is op en (b y regularit y) and admissible (b y saturation). A gts X will be called wea kly discrete if all its singlen tons are op en subsets , discrete if a ll its subsets are op en, and top ological discrete if all families of subsets of X are op en and admissible . (F or a top ological discrete space X , the form ula X = L x ∈ X { x } applie s. All small topological disc rete spaces are finite.) The space from Example 3.15 is w eakly disc rete but not discrete. Example 3.21. Th e semialgebr aic sp ac e R has a smal l infinite we akly dis- cr ete s ubsp ac e N . T he subsp ac e N differs fr om t he sp a c e fr om Example 3.15. Example 3.22 (infinite discrete small spaces) . On any in fi nite set X , ther e is stil l a gener aliz e d t op olo gy making X a discr ete smal l sp ac e. It is enough to set: op en subset is a ny subset, admissible fami l y is any essential ly finite family. Example 3.23. T h e iden t ity mappin g fr om the infinite discr ete s m al l sp ac e on some (infini te) set X to the top olo gic al discr ete sp a c e on this set is a close d a n d op en strictly c ontinuous b ije ction , but not a strict home omo r- phism. A gts X will b e called connected if X there is no pair U, V of op en, disjoin t , nonempt y subsets of X suc h that U ∪ V = X . Prop osition 3.24. If X is a smal l sp ac e, x ∈ X , a nd { C α } α ∈ A is a family of c onne cte d subsets o f X e ach c ontaining x , then S α ∈ A C α is c onne cte d. Pr o of. Assu me that the open subsets U, V of S α ∈ A C α are disjoint and nonempt y , co v er S α ∈ A C α , and x ∈ U . Let y ∈ V . Then y ∈ C α 0 for some α 0 ∈ A . The se ts U ∩ C α 0 and V ∩ C α 0 are open, nonempt y , and co ve r C α 0 , th us C α 0 is not connected, con tradiction. Hence S α ∈ A C α is conne cted. The connected comp onen t of a p oin t x ∈ X of a small space X is the largest connected set C x con taining x . Since the w eak closure of C x is connected, C x is w eakly closed . The quasi-componen t ˜ C x of x is the in tersec tion of clop en subsets of X con taining x . Eac h quasi-comp onen t is also weak ly closed. Eac h connec ted comp onen t is con tained in a quasi - comp onen t. W e will sa y that a space X satisfies ( ACC ) if the family of connec ted comp onen ts of X is op en and admiss ible. Prop osition 3.25. Each smal l sp ac e satisfying ( AC C ) has a finite numb er of c onne cte d c omp onents. Pr o of. Choose one p oin t in an y connected comp onen t of the space. The resulting subspace is top ological disc rete and small, so it is finite. 12 4 Lo cally small s paces In this c hapter w e rebuild the theory of lo cally semialgebraic spaces from [3] on a pure top ological lev el. A gts is lo cally small if there is an admissible cov ering of the whole space b y small o pen subsets. In o t her w ords: a gts X is lo cally small iff X = a S S mO p X . Lo cally small gts es f o rm a full sub category LSS of GTS . Notice that in a lo cally small gts eac h small set is containe d in a small op en set. A family of subsets of a lo cally small space is called lo cally finite if eac h op en small se t meets only finitely man y mem b ers of the family . Example 4.1. Each top olo gic al sp ac e ( X , τ ) c a n b e c onsider e d as a lo c al ly smal l sp ac e r elative to a chosen op en c overin g U of the whole of X i n the fol- lowing way: we de c l a r e memb ers of U to b e smal l sp ac e s , and the gen e r alize d top olo gy on X is then given by the f ormula X = a S U . In p articular, e ach top olo gi c al discr ete sp ac e is lo c al ly smal l ( a nd it i s not smal l, if infini te). Notice that T -spaces of [4] are (practically) lo cally small spaces in the ab o ve sense, whic h is the purp ose o f introducing the family T loc and the Grothendiec k site X T loc in [4]. F or small spaces of ty p e R n sth , with the underlying set R n , w e will consider their lo calization in the followin g sense: assume that eac h of t he op en balls B n cen tered at the orig in with radius n ∈ N \ { 0 } is op en, and consider the lo calization ( R n sth ) loc to b e the admissible uni on of this family of op en small balls. Example 4.2. The sp ac es R n san , R n suban fr om Example 3.1 a s wel l as the lo c al i z ations ( R n salg ) loc , ( R n ts ) loc ar e lo c al ly s mal l, but not smal l. Prop osition 4.3. An op en family of a lo c al l y smal l sp ac e is admissible if and on l y if it is “ lo c al ly essential ly finite ”, which me an s: the family is essential ly fin i te on e ach smal l o p en subset. Pr o of. Eac h admissible family is lo cally essen tially finite by the definition of a small subset. If an op en family V is lo cally esse n tially finite, then it is ess en t ia lly finite on mem b ers of an admiss ible co v ering U o f t he space by op en small subspaces . It means that for eac h member U of U , the family U ∩ V is a dm issible. By the transitivit y axiom, the family U ∩V is admissible. By the saturation axiom, the family V is admissible. Corollary 4.4. Ea c h lo c al ly finite o p en fami l y in a lo c al ly smal l sp ac e is admissible. Prop osition 4.5. Each lo c al ly essential ly finite union of close d subsets of a lo c al ly smal l sp ac e is close d. Pr o of. Let Z b e a lo cally essen tially finite union of closed sets. T ak e an admissible cov ering U of the space b y small open su bsets. F o r eac h elemen t U of U , the set Z ∩ U is a finite union of relativ ely closed subsets of U , so it is relativ ely closed. Eac h relative complemen t Z c ∩ U is op e n in U . By regularit y , the set Z c is op en. 13 Lemma 4.6. The c a t e gory LSS has finite pr o ducts. Pr o of. F or a f amily X 1 , ..., X k of lo cally small spaces a ssume t hat their admissible co verigs by small op en subsets U 1 , ..., U k , res p ectiv ely , are giv en. Then the pro duct spac e is giv en b y the form ula X 1 × ... × X k = a [ U 1 × ... ×U k . The pro jections π i : X 1 × ... × X k → X i are obv iously strictly contin uous, and for giv en strictly contin uous f i : Y → X i , the mapping ( f 1 , ..., f k ) : Y → X 1 × ... × X k is also strictly contin uous, since “admissible” means “lo cally ess en tially finite”. The next example shows that the canonical pro jections along lo cally small spaces ma y not b e op en. Example 4.7. Consider the pr oj e ction π 2 : R loc × R → R ( a s lo c al ly semial- gebr aic sp ac es, se e [3] o r [11]). The image of an op en lo c al ly semialgebr aic set S n ∈ N ( n, n + 1) × ( n, n + 1) under π 2 is not an op en subsp ac e of R . F act 4.8. If U = { U i } i ∈ I is a l o c al ly fin it e op en family in some lo c al ly smal l sp ac e, and for e ach i ∈ I some op en V i ⊆ U i is given, then { V i } i ∈ I is lo c al ly finite, thus admissible. F act 4.9. If X is a smal l sp ac e and Y is a lo c al ly smal l sp ac e, then the pr oje c tion X × Y → Y i s an op en and close d mapping. On lo c ally small spaces w e introduce a top ology , called the strong top ology , whose basis is the family of op en sets of the gts . (Th e mem- b ers of the strong top ology are exactly w eakly op en subsets of the space.) P assing to the strong top ology fo rms a functor () top : LSS → T op . A lo cally small space is called: paracompact if there is a lo cally finite co v ering of the space b y small op en subsets, and Lindel¨ of if there is a coun table admissible cov ering o f the space by small op en subsets . Eac h connected paracompact lo cally small space is Lindel¨ of (the pro of of I.4.17 in [3] is purely top ological). Prop osition 4.10 (cf. I.4.6 of [3]) . F or e ach p a r ac omp ac t lo c al ly smal l sp ac e X , the we ak closur e Y (that is: the closur e in the str ong top o l o gy) o f a smal l set Y is smal l. Pr o of. T ak e a lo cally finite co v ering U of X b y small o pen subsets. The s et Y is co v ered b y a finite sub co ve r U 0 of U of all mem bers of U that meet Y . Then Y and the union of U \ U 0 are disjoin t, and Y is containe d in the union of U 0 , whic h is a small set. Example 4.11. I f a metric sp ac e ( X , d ) satisfies the b al l p r op erty ( BP ) e ach interse ction of two op en b al ls is a finite union of op en b al ls, then X has a na tur a l gener alize d top olo gy, wher e an op en set Y is such a subset of X that the tr ac e of Y on e ach op en b al l is a finite union of 14 op en b al ls, and the adm i ssible c overings ar e such o p en c overings that ar e essential ly finite on e ach op en b al l. Then op e n b al ls ar e smal l sets. The c overing of the sp a c e by a l l op en b al ls is admis s i ble (e ach smal l set is c over e d by one o p en b al l), and X is a lo c al ly sm al l sp ac e sa t isfying the first axiom of c ountability. The family of al l op en b al ls is a b asi s of the gener a lize d top olo gy. A subs et Y of a lo cally small space X is locally constructible if eac h in tersec tion Y ∩ U with a small op en U ⊆ X is constructible in U (so also in X ). The Bo olean algebra of lo cally constructible subsets of a lo cally small space may b e strictly larger than the Bo olean algebra of constructible subsets ( to se e this one can construc t a sequen ce X n of constructible sub- sets of some small spaces Z n eac h X n needing at least n op en sets in the description, and then glue the spaces Z n in to one lo cally small space). W e will sa y that a locally small space has the closure property if the follo wing holds: ( CP ) the w eak closure of a small lo cally closed subset is a closed subset. F act 4.12. Each lo c al ly smal l sp ac e with a r e gular str ong top olo gy and with the closur e pr op erty is we akly r e gular. F act 4.13. Each lo c al ly c ons tructible subse t of a lo c al ly sma l l sp ac e is a lo c al ly essential ly finite unio n of sm al l lo c al ly cl o se d subsets. Prop osition 4.14. I f a lo c al ly smal l sp ac e has the closur e pr op erty, then the we ak closur e o f e a ch lo c al ly c onstructible s e t is a clos e d set. Pr o of. By the F act 4.13 , the w eak clos ure o f eac h lo cally constructible s et is a union of weak closures of some lo cally essen tially finite family of small constructible sets. But the family of these closures is also lo cally essen tially finite, and the thesis follo ws from Proposition 4.5 . By the ab o v e, if a lo cally small space has the closure prop ert y , then the closure op erator o f the generated to p ology restricted to the class of lo cally constructible sets ma y b e considered as the c l o s u r e op er ator of the generalized top ology (in g e neral, the closure op erator on a gts do es not exist). Notice that eac h subset Y of a lo cally small space X forms a (lo cally small) subspace, since if X = a S α X α with all X α small op en, then the form ula Y = a S α ( X α ∩ Y ) defines a lo cally small space that is a subspace of X . Hence Prop osition 3.2 4 , the concept of a connected comp onen t, and a quasi-comp onen t extend to the category LSS . Also the equalizers for parallel pairs of morphisms exist in LSS . T ogether with Lemma 4.6 this giv es Theorem 4.15. The c ate gory LSS is finitely c om plete. 15 5 W eakly small s paces No w we start to rein tro duce the theory of w eakly semialgebraic spaces from [7] on a top ological lev el. A weakly (or piecewise ) small spa ce is a gts X ha ving a family ( X α ) α ∈ A of closed small subspaces indexed by a partia lly ordered set A suc h that the followi ng conditions hold: W1) X is the union of all X α ’s as sets , W2) if α ≤ β then X α is a (closed, small) subspace of X β , W3) fo r eac h α ∈ A there are only finitely man y β ∈ A suc h that β < α , W4) for eac h t w o α, β ∈ A there is γ ∈ A suc h that X α ∩ X β = X γ , W5) for eac h t w o α, β ∈ A there is γ ∈ A suc h that γ ≥ α and γ ≥ β , W6) the g ts X is the inductiv e limit of the directed family ( X α ) α ∈ A , whic h means: a) a subset U of X is op en iff all sets X α ∩ U are op en in resp ectiv e X α s, b) an op en family U is admissible iff all for eac h α ∈ A the fa m ily U ∩ X α is admissible (=essen tially finite) in resp ec tiv e X α . Suc h a family ( X α ) α ∈ A is called an exhaustion of X . The w eakly small spaces form a full sub category WSS of GTS . Here “admissible” means “piecewis e essen tially finite”, where “piecewise ” means “when restricted to a mem b er of the exhaustion (c hosen to witness that X is a w eakly small space)”. Mem b ers of this exhaustion of X ma y b e called pieces . If ( X α ) α ∈ A is an exhaustion of X , then w e will write X = e S α ∈ A X α . The index function for the exhaustion ( X α ) α ∈ A is the function η : X → A giv en b y the form ula η ( x ) = inf { α ∈ A | x ∈ X α } . Here infim um exists thanks to W3). The index function η giv es a decom- p osition of the space X into sm all lo cally closed subspaces X 0 α = η − 1 ( α ) = X α \ S β <α X β . F act 5.1. A subset Y of a we akly sm al l sp ac e X is close d if and on ly if it is pie c ewise close d. Prop osition 5.2. A pie c ewise essential ly finite union of close d subsets of a we akly sm a l l sp ac e is close d . Pr o of. An essen tially finite union of closed subsets is closed. Hence a pie ce- wise essen tially finite union o f closed sets is piecewis e closed, thus closed. Notice that the c hosen exhaustion of a w eakly small space is a piecewise essen tially finite (relativ e to this exhaustion) family of closed sets, and re- mind that a cons tructible subset o f a pie ce is a fin ite un ion of lo cally closed subsets o f a piece. A weakly (or piecewise ) constructible subset is suc h a subset Y ⊆ X that has constructible in tersections with all mem b ers of the c hosen exhaustion ( X α ) α ∈ A . Prop osition 5.3. Pie c ewise c onstructible s u bsets of a we akly smal l sp ac e ar e ex a ctly p ie c ewise essential ly finite unions of lo c al ly close d subsets of pie c e s. 16 Pr o of. Piec ewise constructible subsets are piecewise essen tially finite unions of lo cally closed subsets of pieces. An essen tially finite union of lo cally closed subsets of pieces is a finite union of lo cally closed subsets of a single piece, so a constructible subset of a piece . Now apply “piece wise”. The strong topology on X = e S α ∈ A X α is the top ology that mak es the top ological space X t he respectiv e inductiv e limit of the system of top o- logical sp aces X α . Its mem b ers are all the piecewise w eakly op en subsets, not only the w eakly (piecewise) o p en subsets. Hence the op en sets from the generalized topolog y ma y not form a basis of the strong top ology (see App en dix C of [7]). Another unpleasan t fact ab out the w eakly small spaces (comparising with the lo cally small spaces) is that p oin ts ma y not ha v e small neighborho o ds (consider an infinite w edge of circles as in Example 4.1.8 of [7]). W e will say that a w eakly small space has the closure prop ert y if the follo wing holds: ( CP ) the w eak closure of a lo cally closed sub set of a piece is a close d subset. Notice that the strong topology and the generated top o logy coincide on ev ery piece. If the space X has the closure prop ert y , then the top ological closure op erator restricted to the class of constructible subsets of pieces of X ma y be treated as the closur e op er ator of the generalized top ology . Notice that eac h subset Y of a we akly small space X forms a (w eakly small) subspace , since if X = e S α X α with all X α small closed, then Y = e S α ( X α ∩ Y ) defines a we akly small space that is a subspace of X . Hence Prop o- sition 3.24, the concept of a connected comp onen t, and a quasi-componen t extend to the category WSS . Theorem 5.4 (cf. IV.2.1 in [7]) . If a we akl y smal l sp ac e X = e S α ∈ A X α is str ong ly T 1 , a n d L is a smal l sp ac e, then for e ach s trictly c o n t inuous mapping f : L → X ther e is α 0 ∈ A such that f ( L ) ⊆ X α 0 . Pr o of. Let η : X → A denote the index function for the exhaustion ( X α ) α ∈ A . If η ( f ( L )) w ere infinite, then f o r each α ∈ η ( f ( L )) w e could c hoo s e x α ∈ f ( L ) with η ( x ) = α and some y α ∈ f − 1 ( x α ) . Set S = { x α : α ∈ η ( f ( L )) } . F or eac h γ ∈ A , the se t S ∩ X γ is finite. Since X is strongly T 1 , the set S is closed as w ell as eac h of its subsets, so S is to pological discrete. Then { y α : α ∈ η ( f ( L )) } ⊆ f − 1 ( S ) is a top o logical discrete, infinite, and small subset of L . This is a con tradiction. Hence η ( f ( L )) is finite, a nd there is α 0 ≥ η ( f ( L )) . W e get f ( L ) ⊆ X α 0 . Theorem 5.5 (cf. IV.2.2 in [7]) . If a we akly sm a l l sp ac e X with an exhaus- tion ( X α ) α ∈ A is str ong ly T 1 , then e ach smal l s ubsp ac e L of X is c ontaine d in some X α 0 . In p articular, e ach memb er X β of any exhaustion ( X β ) β ∈ B of X is c o n t aine d in some memb er X α 0 of the ini tial exhaustion. 17 Pr o of. F or eac h s uc h L , the inclus ion mapping i : L → X is strictly contin- uous. By Theorem 5 .4 , the se t i ( L ) = L is con tained in a mem ber of the exhaustion ( X α ) α ∈ A . Let us denote by WSS 1 the full subcategory of WSS comp osed of strongly T 1 ob jects of WSS . In this category the term “piecewise” do es not dep end on an ex haustion (as it ma y b e expressed b y “when restricted to a closed small set”), hence passing to the strong top ology forms a functor () stop : WSS 1 → T op . Theorem 5.6. Th e c ate gory WSS 1 is finitely c omplete. Pr o of. It is enough to consider binary pro ducts. If X , Y ha v e exhaustions ( X α ) α ∈ A , ( Y β ) β ∈ B , resp ectiv ely , then ( X α × Y β ) ( α,β ) ∈ A × B is a n exhaustion defining the w eakly small space X × Y . Notice that X × Y is strongly T 1 . The pro jections are ob viously strictly con tin uous, and for strictly con tin uous f : Z → X , g : Z → Y , the mapping ( f , g ) : Z → X × Y is strictly con tin uous, since if Z = e S γ ∈ Γ Z γ , then for each Z γ 0 there are X α 0 and Y β 0 suc h that f ( Z γ 0 ) ⊆ X α 0 and g ( Z γ 0 ) ⊆ Y β 0 . Thu s it is the pro duct of X and Y in WSS 1 . Since each sub set forms a subspace, the ex istence of eq ualizers for pairs of parallel mappings is clear. F act 5.7. If X is a smal l sp ac e and Y is an obje ct of WSS 1 , then the pr oje c tion X × Y → Y i s an op en and close d mapping. 6 Spaces o v er structure s In this ch apter, w e deal with lo cally definable and w eakly definable spaces o v er mathematical structures. Assume that M is any (one sorted, first order for simplicit y) structure in the sense of mo de l theory . A function sheaf o v er M on a gts X is a sheaf F of sets on X (the sheaf property is assumed o nl y f or admissible co v erings) suc h that for eac h o pen U the se t F ( U ) is con tained in the set M U of a ll functions from U into M , and the restrictions o f the sheaf are the set-theoretical restrictions o f functions. A space o v er M is a pair ( X , O X ) , where X is a gts and O X is a function sheaf ov er M on X . A morphism f : ( X , O X ) → ( Y , O Y ) of spaces o v er M is a strictly con tinuous mapping f : X → Y suc h that for eac h op en subset V of Y the set-theoretical substitution h 7→ h ◦ f gives the mapping f # V : O Y ( V ) → O X ( f − 1 ( V )) . (W e could informally say that f # : O Y → O X is the “morphism of function shea ves ” ov er M induced set-theoretically by f . W e can also define for function shea v es ( f ∗ O X )( V ) = { h : V → R | h ◦ f ∈ O X ( f − 1 ( V )) } , and then eac h f # V : O Y ( V ) → f ∗ O X ( V ) is an in clusion.) An isomorphism is an in v ertible morphism. W e get a category Space ( M ) of spaces o ve r M and their morphisms. 18 Theorem 6.1 (cf. I.2, p. 11, in [3 ]) . F or e ach structur e M , the c ate gory Space ( M ) is c o c omplete. Pr o of. Assu me that X is the inductiv e limit (in the category GTS ) of a diagram D with ob jects ( X i ) i ∈ I , indexed b y a small category I , and the canonical morphisms φ i : X i → X . Assume additionally that function sheav es O X i o v er M a r e giv en. Then define O X ( U ) = { h : U → M | ∀ i ∈ I h ◦ φ i ∈ O X i ( φ − 1 i ( U )) } . Since all O X i are f unction sheav es and the section amalgamation in function shea ves is giv en b y the union of g r a ph s of sections, also O X is a function sheaf. By the ab o v e definition, eac h φ i is a morphism of spaces o ver M . If another space ( Y , O Y ) with a set o f morphis ms ψ i : X i → Y is giv en, then there is a unique morphism η : X → Y in GTS suc h that ψ i = η ◦ φ i for a ll i . But η is also a morphism in Space ( M ) , b ecause t he condition k ◦ η ∈ O X ( η − 1 ( W )) for each k ∈ O Y ( W ) is visibly satisfied, since , for eac h i ∈ I , w e ha v e k ◦ η ◦ φ i = k ◦ ψ i ∈ O X i ( ψ − 1 i ( W )) = O X i ( φ − 1 i ( η − 1 ( W ))) . No w assume that a top ology is giv en on the underlying set M of a (firs t order) structure M . W e explicitely demand that the pro duct top ologies on cartesian p o w ers M n should b e considered. W e will call suc h M a w eakly top ological structure . (This setting seems to coincide with the case (i) in the in tro duction of [12]. W e do not explore a sp ecial language L t for top ological structures considered in the case (ii) of the introduction of [12] or in [5].) Then all pro jections π n,i : M n → M n − 1 (forgetting the i -th co ordinate) are contin uous and op en. Th us, for example, the field of complex n um b ers ( C , + , · ) considered with the euclidean top ology (but not with the Z a ris ki to pology) is a we akly top ological structure. Also the fields Q p of p-adic num b ers considered with their natural top ologies (coming from v aluations) are w eakly top ological. No w for eac h definable set D ⊆ M n , w e set (as in [11]): a) an op en subset of a gts means a relativ ely op en, defin able subset; b) an admissible fami l y of a gts means an essen tially finite family . Eac h suc h D b ecomes a small gts , and π n,i b ecome strictly con tin uous and op en. W e define the structure sheaf O D as the sheaf of all definable con tin uous f unctions from resp ec tiv e ( gts -)op en subsets U ⊆ D into M . Th us ( D , O D ) b ecomes a sp ace ov er M . An op en subspace of a space ov er M is an op en subset of its g t s together with the function sheaf of the space restricted to this op en set. A small subspace K of a space X o v er M is a small sub set of its gts with the function sheaf O K comp ose d of all finite op en unions of restrictions of sections of the function sheaf O X to the relativ ely op en subsets of the small subspace K . 19 An affin e definabl e space o ver M is a space o v er M isomorphic to a definable subset of some M n considered with its usual structure of a space o v er M . A definabl e space ov er M is a space ov er M that has a finite op en co v ering b y affine de finable subspaces. (The stru cture sheaf is determine d in a n obvious w a y .) A lo cally definable space ov er M is a space o v er M that has an a dm issible cov ering by affine definable op en subspaces . (The sections of a structure sheaf are admissible unions o f se ctions of the structure shea ves o f affine op en subspaces.) A w eakly definable space X o v er M is a space o ve r M that has an exh austion ( X α ) α ∈ A comp ose d of definable (small) subspace s such that a func tion h : V → M is a section of O X iff all restrictions h | V ∩ X α are sections of respectiv e O X α s. Morphisms of affine definable spaces, definable spaces, lo cally defin- able spaces, and w eakly definable spaces ov er M are their morphisms as spaces o v er M . W e g e t full subcategories ADS ( M ) , DS ( M ) , LDS ( M ) , WDS ( M ) of Space ( M ) . Notice that eac h definable subset of an affine definable space is also an affine definable space. Th us definable subsets of definable spaces are natu- rally subsp aces in D S ( M ) . Lo cally definable sub sets of lo cally definable spaces are naturally subspaces in LDS ( M ) . Also piecewis e definable sub- sets o f w eakly definable spaces are naturally subspa ces in WDS ( M ) . It is clear that ADS ( M ) , DS ( M ) , LDS ( M ) , WDS ( M ) ha v e equalizers for pairs of parallel morphisms . The notions of paracompactness and Lindel¨ ofness for lo cally defin- able spaces coincide with their coun terparts for the underlying lo cally small spaces. As in [11], w e get from the definitions F acts 6.2 (F acts 5 in [11]) . D efinable sp ac es ar e smal l. Every smal l subsp ac e of a lo c al ly de fi nable sp ac e is definable. Notice that if M is T 1 , then all ob jects of LDS ( M ) and WDS ( M ) are strongly T 1 . Moreo v er, each w eakly T 1 lo cally definable or w eakly definable space o v er M is strongly T 1 . (It is visible for affine spaces, and extends to the general case by applying “lo cally” or “piecewis e”.) Th us w e can sp eak just ab out T 1 ob jects of LDS ( M ) or of WDS ( M ) for any M . F rom Theorem 5.4, w e get F act 6.3. If a we akly defin able sp ac e is T 1 , then e ach smal l subsp a c e is c ontaine d in a pie c e, he n c e it is a definable subsp ac e. Theorem 6.4. F or e ach we akly top olo gic al structur e M , the pr o ducts of smal l sp a c es in the c ate gory Space ( M ) exist. Pr o of. Let a family { ( X α , O X α ) } α ∈ A of small spaces o v er M b e giv en. By the Prop osition 3.16, the family { X α } α ∈ A has the pro duct X in GTS . Now , for any h ∈ O X α ( V ) , t he f unction h ◦ π α : π − 1 α ( V ) → M may be called a “function of one v ariable”, where π α : X → X α is the pro jection. D e fine sections of O X as all admissible (so essen tially finite) unions of compati- ble functions of o ne v ariable. Then the pro jections π α are morphisms of 20 Space ( M ) . F or a family f α : Y → X α of morphisms of Space ( M ) , the induced mapping ( f α ) α ∈ A : Y → X is a morphism of Space ( M ) , since all the sections of O X are finite unions of the functions of one v ariable, whic h are determined b y sections of O X α s. Unfortunately , ev en finite pro ducts of affine definable spaces in Space ( M ) usually are not affine definable spaces . Prop osition 6.5. F or e ach we akly top olo gic al structur e M , the c ate gories ADS ( M ) and DS ( M ) ar e finitely c om plete. Pr o of. W e need only to c hec k if finite pro ducts exist in ADS ( M ) and DS ( M ) . F or definable sets D ⊆ M n and E ⊆ M m , the pro duct is the definable set D × E ⊆ M n + m (with its natural space structure). Indeed, the pro jections a re c on tin uous definable mappings, and for any con tin uous definable f : Z → D , g : Z → E , with Z definable in some M k , the induced mapping ( f , g ) : Z → D × E is contin uous definable, since the top ology of D × E is the pro duct top ology . F or a contin uous definable f unction h ∈ O D × E ( U ) , the function h ◦ ( f , g ) : ( f , g ) − 1 ( U ) → M is contin uous de- finable. This finishes the pro of f or the affine definable case. The definable case is similar. Notice that the canonical pro jections from a finite product of affine definable spaces to its factors are op en morphisms, but not in general closed morphisms. F act 6.6 (cf. I.1.3 in [3]) . A mapp ing f : X → Y b etwe en obje cts of LDS ( M ) is a m orphism iff the imag e of e a c h op en definable subsp ac e is c ontaine d in an op en definable subsp ac e and when r estricte d to op en defin- able subsp ac es in the domain and in the r ange, the m apping is c ontinuous definable. Theorem 6.7 (cf. I.2.5 in [3]) . F or e ach we akly top olo gi c al structur e M , the c ate gory LDS ( M ) is finitely c omplete. Pr o of. W e need only to chec k the existence of finite pro ducts. F or lo cally de- finable spaces ( X , O X ) = a S α ∈ A ( X α , O X α ) , ( Y , O Y ) = a S β ∈ B ( Y β , O Y β ) , where ( X α , O X α ) , ( Y β , O Y β ) a re affine definable subspaces ov er M , consider the space a [ ( α,β ) ∈ A × B ( X α , O X α ) × ( Y β , O Y β ) = a [ ( α,β ) ∈ A × B ( X α × Y β , O X α × Y β ) . This is the pro duct o f given lo cally definable spaces , since the pro jections are visibly morphisms in LDS ( M ) , and for giv en morphisms f : Z → X , g : Z → Y in LDS ( M ) from the space ( Z , O Z ) = a S γ ∈ Γ ( Z γ , O Z γ ) , the induced mapping ( f , g ) : Z → X × Y is a morphism in LDS ( M ) . Indeed, w e ma y assume { Z γ } γ ∈ Γ is an admissible cov ering b y affine definable spaces that is a refinemen t of the preimage of the co v ering { X α × Y β } ( α,β ) ∈ A × B and thus eac h Z γ is mapp ed in to some X α × Y β b y some contin uous definable mapping, and the whole ( f , g ) is an admissible union of suc h partial mappings. 21 W e can also generalize the follo wing concept from [11]: a subset Y of a lo cally definable sp ace X (o v er M ) may b e called lo cal if f or each y ∈ Y there is an op en small neigh b orho o d U of y suc h that Y ∩ U is definable in U . On suc h Y , w e could define a lo cally defin able space b y the form ula Y = a [ { Y ∩ U | U small, op en , Y ∩ U definable in U } . (This definition is canonical in the sense that it do es not depend o n an y arbitrary c hoice of op en neigh b o rhoo ds.) How ev er, the use of lo cal subsets do es not reflect this lo cally definable space (see Examples 11 and 12 in [11]). An y w eakly op en and a n y w eakly discrete set is lo cal. Let us denote b y WDS 1 ( M ) the full subcategory of T 1 ob jects of WDS ( M ) for an y M . F act 6.8 (cf. IV.2.3 in [7]) . A map ping f : X → Y b etwe en obje cts of WDS 1 ( M ) is a morphism iff the imag e of a pie c e is c ontaine d a in pie c e and wh e n r estricte d to pie c es in the d omain and in the r ange, the mapping is c ontinuous defin able. Notice that for t wo morphisms f 1 : X 1 → Y 1 , f 2 : X 2 → Y 2 of LDS ( M ) or of WDS 1 ( M ) , their cartesian pro du ct f 1 × f 2 : X 1 × X 2 → Y 1 × Y 2 is a morphism. Theorem 6.9 (cf. IV.3, p. 32, in [7]) . F or e ach we akly top olo gic al structur e M , the c ate gory WDS 1 ( M ) i s finitely c o m plete. Pr o of. W e need only to c hec k the existenc e o f finite pro ducts. Notice that b y Theorem 6.7, finite pro ducts exist in the category DS ( M ) , and t he pro duct of T 1 spaces is T 1 . W e will drop the structure shea v es in not a - tion. I f ( X α ) α ∈ A , ( Y β ) β ∈ B are exhaustions of ob jects X , Y , then define X × Y = e S ( α,β ) ∈ A × B ( X α × Y β ) ( α,β ) ∈ A × B . This is an exhaustion determin- ing the structure sheaf. The resulting space is clearly T 1 , and the pro jec- tions X × Y → X , X × Y → Y a re visibly morphisms o f WDS 1 ( M ) . If Z = e S γ ∈ Γ Z γ is any ob ject of WDS 1 ( M ) and f : Z → X , g : Z → Y are morphisms, then the mapping ( f , g ) : Z → X × Y is a morphism b e- cause: w e ma y assume that for eac h Z γ 0 there are X α 0 and Y β 0 suc h that f ( Z γ 0 ) ⊆ X α 0 and g ( Z γ 0 ) ⊆ Y β 0 , the generalized top ology of the pro duct of definable spaces contains the generalized top ology of their (generalized top ological) pro duct, and the definable con tin uous m appings are stable un- der comp osition and juxtap osition. Th us X × Y is the product of X and Y in WDS 1 ( M ) . Prop osition 6.10 (cf. I.3.5 in [3]) . F or any we ak ly top o lo gi c al structur e M , the c ate gory LDS ( M ) has fib er pr o ducts. Pr o of. The fib er pro duct of morphisms f : X → Z and g : Y → Z is the subspace ( f × g ) − 1 (∆ Z ) ⊆ X × Y , where ∆ Z is the diagonal of the space Z (lo cally defin able in Z × Z ). 22 Prop osition 6.11 (cf. IV.3.20 in [7]) . F or any we akly top olo gic al s tructur e M , the c ate gory WDS 1 ( M ) h as fib e r pr o ducts. The pro of of this pro p osition is similar to the pro of of the previous prop osition. Let C b e one of the categories: LDS ( M ) or W DS 1 ( M ) . W e will sa y that an ob ject Z of C is C -complete if the mapping Z → {∗} is univ ersally closed in C , whic h means that fo r eac h ob ject Y of C the pro jection Z × Y → Y (whic h is the base extension of Z → {∗} ) is a closed mapping. Prop osition 6.12. A close d subsp ac e of a C -c o m plete sp ac e is C -c omplete. Pr o of. Let C b e a closed subspace of a C -complete space Z . F or an y Y in C , the space C × Y is a closed subset of Z × Y . Th us the image under the pro jection a lo ng C o f an y closed subset of C × Y is closed in Y . Prop osition 6.13. Every we akly close d subsp ac e of an o b je ct of C i s a close d subsp ac e. Pr o of. This is clear for affine definable spaces. In general, closedness may b e c hec k ed “lo cally” or “piecewise”. Prop osition 6.14. The image g ( C ) of a C -c om plete subsp ac e C of Y under a morphism g : Y → Z in C is a C - c omplete subsp ac e. Pr o of. Consid er an ob ject W of C . F or a closed subset A o f g ( C ) × W , w e ha v e π g ( C ) ( A ) = π C (( g | C × id W ) − 1 ( A )) is a closed set. Prop osition 6.15. F or an obje ct Z of C , the fol lowing c ond i tion s ar e e q u iv- alent: a) Z is we akl y Hausdorff; b) Z is str ongly Hausdorff; c) Z has i ts d i a gonal ∆ Z close d. Pr o of. If a space X is w eakly Hausdorff, then its generated top ology is Hausdorff, th us the diagonal ∆ X is closed in the generated top ology . But ∆ X is alw a ys a subspac e, th us, b y Proposition 6.1 3, a closed subspace. If the diago nal ∆ X of a space X is a closed subspace, then the generated top ology of X is Hausdorff, so X is w eakly Hausdorff. But it is also strongly T 1 , since for any x 0 ∈ X , the set { ( x 0 , x 0 ) } is closed in { x 0 } × X and the pro jection { x 0 } × X → X is closed. Th us X is strongly Hausdorff. Notice that if the top ology o f M is Hausdorff, then: a) all M n as w ell as all ob jects of ADS ( M ) are (strongly) Hausdorff; b) a ll ob jects of W DS ( M ) ha ving exhaus tions consisting of affine de- finable spaces are (strongly) Hausdorff; c) an ob ject of DS ( M ) ma y not b e (w eakly) Hausdorff. Because of 6.15, w e ma y speak just ab out Hausdorff ob jects of C . Prop osition 6.16. I f Y is a Hausdorff obje ct of C , then: a) e ach of its C -c o m plete subsp ac e s is close d; b) the gr aph of e ach morphis m f : X → Y in C is c l o se d; 23 Pr o of. a) If C is a C -complete subspace of Y , then ∆ Y ∩ ( C × Y ) is relativ ely closed in C × Y , and its pro jection on Y , equal to C , is closed . b) The graph of f is a su bspace of X × Y b eing the preimage of ∆ Y b y a morphism f × id Y . Supp ose M satisfies the condition ( DCCD ) M admits (finite) cell decomposition with definably connected cells (here w e assume only that a cell is a definable set, for stronger notions of a cell see [10] or [13]), then: a) each ob j ect of DS ( M ) has a finite nu m b er of clop en definable con- nected comp onen ts, and is a finite (generalized) top ological direct sum of them; b) eac h ob ject of LDS ( M ) is a (lo cally finite) direct sum of its connected comp onen ts; c) eac h ob ject o f WDS ( M ) is a piecewise finite union o f its clop en connected comp onen ts, so a ls o a direct sum of its connected components . Th us If M satisfies ( D C C D ) , then a ll ob j ects of LDS ( M ) and of WDS ( M ) satisfy ( AC C ) . All o-minimal structures M (with their or- der dense) satisfy ( D C C D ) , but t-minimality in the sense of [13] do es not guaran tee ( D C C D ) . Lemma 6.17. If C has a T 1 obje ct having a n o n-close d c ountable set, then a top olo gic al dis cr ete infinite sp ac e is not C -c om p lete. Pr o of. Let Z b e an infinite discrete space, and W a T 1 ob ject with a non- closed coun table set C = { c n } n ∈ N (giv en b y an injectiv e se quence). T ake an injectiv e sequence { z n } n ∈ N of elemen ts of Z . The n C = π Z ( { ( z n , c n ) : n ∈ N } ) , so Z is not complete. Theorem 6.18 (cf. I.5.8 in [3]) . If ther e is a T 1 obje ct Z of C having a non-close d c ountable subset, then e ach C -c o mplete sp ac e interse cts onl y a finite numb er of c o mp o nents of a ge n er aliz e d top olo g ic al sum. Pr o of. The C - complete space C is a direct sum of the family of its connected comp onen ts. T ake one p oin t from each connected comp onen t. The resulting subspace S is discrete and closed, so complete b y Prop osition 6.12. If it we re infinite, then w e could assume (b y taking a closed subspace if nece ssary) it is coun table. An injection from S to a countable non-closed subset of some Hausdorff ob ject of C w ould giv e a coun terexample o f c) in Prop osition 6.16. Hence S is finite. Theorem 6.19 (cf. I.5.10 in [3]) . Assume that ther e is a T 1 obje ct Z of C h aving a non-close d c o untable subset. L et C b e a C -c omplete sp ac e. A dditional ly, i f C = LDS ( M ) then C is assume d to b e T 1 and Lindel¨ of. Then C is a defi nable sp ac e. Pr o of. In the case of a lo cally definable space: If C = a S n ∈ N C n is a n a dm isi- ble co v ering by (affine) definable spaces and C is not definable, then we may 24 assume that for eac h n ∈ N w e can c ho ose x n ∈ C n \ ( C 0 ∪ ... ∪ C n − 1 ) 6 = ∅ . The set B = { x n | n ∈ N } is infinite, not small, lo cally finite subspace, so, as a T 1 space, top ological discrete. Moreov er, B is closed, th us complete (b y Prop osition 6.12). In the case o f a w eakly definable space: if C is not definable, then the index function η o f an exhaustion ( C α ) α ∈ A has infinite image, and w e can c ho o s e an elemen t x α ∈ C 0 α for eac h α ∈ η ( C ) . The set B = { x α : α ∈ η ( C ) } is an infinite (but piecew ise finite) closed subspace all of whose subse ts are also closed. Th us B is a complete (b y Prop osition 6.12) and top ological discrete space. W e ma y assume B is countable . W e get a con tradiction with Lemm a 6.17. Remind (after [12]) that a structure M is a first order top ological structure (called in [5] a top ological structure with explicitely definable top ology) if the basis of the top ology o n M is uniformly definable in M . (There is a form ula Φ( x , ¯ y ) of the (first order) languag e o f M suc h that the family { Φ( x, ¯ a ) M : ¯ a ⊆ M } is the basis of the top ology of M .) If M is a first order top ological structure , then the relativ e closure of a definable subset in a definable set is definable (see [1 2 ]). Th us the closure op erator for the generalized top ology exists for the lo cally definable subsets of lo cally definable spaces, and for definable subsets of pieces in we akly definable spaces ov er M . 7 Op en p roblems Generalized top ology in the sense o f H. D elfs and M. Knebusc h is a new c hapter in general topo logy . That is why there are many unansw ered ques- tions in this topic. The follo wing questions are suggested to the reade r: 1) D oes an y subset of a gts form a subspace? What kinds of subsets (other than the op en subsets and the small subsets ) alw a ys form subspaces? F or what classes of generalized top ological spaces (other than lo cally small or w eakly small) all subsets form subspac es? 2) Do es GTS hav e pro ducts? 3) F or a family { V i } i ∈ I of subsets, and a family {U j } j ∈ J of families of a set X , descri b e the generalized top ology the y generate. Ac kno wledgeme n ts. This pap er w as mainly written during my stay at the F ields Institute du ring the Thematic Program on o-minimal Structure s and Real Analytic Geometry in 2009. I thank the Fields Instit ute for their w arm hospitalit y . References [1] E. Baro, M. Otero, L o c al ly definable homotopy , Annals of Pure a nd Applied Logic 161 (2010), 488–503. 25 [2] M. Barr, C. W ells, T op oses, T riple s an d The orie s , Grundlehren der mathematisc hen Wissensc haft 278, Springer-V erlag 19 8 5. [3] H. Delfs, M. Knebu sc h, L o c al ly Semialgebr aic Sp ac es , Lecture Notes in Mathematics 1173, Springer-V erlag 1985. [4] M. Edm undo, L. Prelli, She ave s on T -top olo gies , [math.LO]. [5] J. Flum, M. Ziegler, T op ol o gic al Mo del The ory , Lecture Notes in Math- ematics 769, Springer-V erlag 1980. [6] M. Kashiw ara , P .Shapira, C a t e gories and She aves , Grundlehren der mathematisc hen Wissensc haften 332 , Springer-V erlag 20 06. [7] M. Knebusc h, We akly Semialgebr a ic Sp ac es , L ecture Notes in Mathe- matics 1367, Springer-V erlag 1989. [8] M. Knebusc h, Sem ialgebr aic top olo gy in the r e c ent ten ye ars , ”Real Algebraic G eometry Pro ceedings, Rennes 1991” , M. Coste, L. Mah ´ e, M.-F. Roy , eds., LNM 1524, Springer 1992, 1–36. [9] S. Mac Lane, I. Mo erdijk, She aves in Ge ometry and L o gic , Univ ersi- text, Springer-V erlag 1992. [10] Mathews, Ce l l de c omp osition an d dimension func tion s in first or der top olo gi c al structur es , Pro c. London Math. So c. (3) 70 (1995), 1–32. [11] A. Pięk osz, O-m i nimal homotopy and ge ner ali z e d (c o)homolo gy , to app ear in Ro c ky Moun tain Journal of Mathematics . [12] A. Pillay , First or der top olo gic al structur es and the o ries , Journal of Sym b olic Logic V o l. 53, No. 3 (1987), 763– 7 78. [13] H. Sc houtens, T-minimality , preprin t, 2001. Politechnika Krak o wska Instytut Ma tema tyki W arsza wska 24 PL-31-155 Krak ó w Poland E-mail: pupieko s@cyf-kr.e du.pl 26