Some Generalizations of Fedorchuk Duality Theorem -- I

Some Generalizations of F edorc h uk Dualit y Theorem – I Georgi Dimo v ∗ Dept. of Math. and Informatics, So fia Universit y , 5 J. Bourchier Blvd., 1164 Sofia, Bulgar ia Abstract Generalizing Duality Theorem of V. V. F edorc hu k [11], we prove Stone-t yp e duality theorems for the following four categor ies: all of them hav e as ob jects the lo c a lly com- pact H ausdorff spaces, and their mor phisms are, re s pe c tively , the contin uo us skeletal maps, the quasi-o p e n per fect maps, the o p e n maps, the op en per fect maps. In pa rti- cular, a Stone-type duality theorem for the category of all compact Hausdor ff spaces and all op en maps b et ween them is prov ed. W e also obtain equiv alence theore ms for these four catego ries. The v ersions of these theorems for the full sub categ ories o f these categories having as ob jects all lo cally compact connected Hausdor ff spa c es are formulated as well. MSC: primar y 5 4D45, 18A40; seco ndary 0 6E15, 54C1 0, 54E 05. Keywor ds: Normal contact algebra; Lo cal contact algebra; Compact spaces; Lo cally compact spaces; Skeletal ma ps; (Q ua si-)Op en per fect maps; Op en maps; Perfect maps; Duality; Equiv a- lence. In tro ducti o n According to the famous Stone Duality Theorem ([22]), the category of all zero- dimensional compact Hausdorff spaces and all contin uous maps b et w een them is dually equiv alen t to the categor y B o ol of all Boolean algebras and all Bo olean homomorphisms betw een the m. In 1962, H. de V ries [4] in tro duced the no t ion of c ompingent Bo ole an algebr a and prov ed tha t the category of all compact Hausdorff spaces and a ll contin uous maps b et w een them is dually equiv alen t t o the category ∗ This pap er was supp orted by the pr o ject MI-1 5 10/2 007 “Applied Logics a nd T opo logical Struc- tures” of the Bulg arian Ministry o f Education and Science. 1 E-mail addr ess: gdimov@fmi.uni-sofia.bg 1 of all complete compingen t Bo o lean algebras a nd appropriate morphis ms b et w een them. Using de V ries ’ Theorem, V. V. F edorch uk [11] sho wed that the category Sk eC of all compact Hausdorff spaces a nd all quasi-op en maps b et w een them is dually equiv alen t to the category DSkeC of all complete normal con tact algebras and all complete Bo olean homomorphisms b et w een them satisfying one simple con- dition (see Theorem 2.13 b elo w). The normal c ontact a l g e br as (briefly , NCAs) are Bo olean algebras with an additio nal relation, called c ontact r elation . The axioms whic h this contact relation satisfies are v ery similar to the axioms of Efremo viˇ c pr ox- imities ([1 0]). The notion o f normal con tact a lg ebra w as in tro duced b y F edorc h uk [11] under t he name Bo ole an δ - algebr a as an equiv alen t expression of the notion of compingen t Bo olean algebra of de V ries . W e call suc h algebras “normal contact alge- bras” b ecause they form a sub class of the class of c ontact algebr as intro duced in [7]. In 1997, Ro ep er [20] defined the notion of r e gion-b ase d top olo gy a s one of the p ossible formalizations of the ideas of De Laguna [3] and Whitehead [2 4] for a region- based theory o f space. F ollowing [23 , 7], the region-based top ologies of Ro ep er app ear here as lo c al c ontact algebr as (briefly , LCAs), b ecause the axioms whic h they satisfy almost coincide with the axioms o f lo cal pro ximities of Leader [14]. In his pap er [20], Ro ep er prov ed the follo wing theorem: there is a bij ective corresp ondence b et w een all (up to homeomorphism) lo cally compact Hausdorff spaces and all (up to isomor- phism) complete LCAs. It generalizes the theorem o f de V ries [4] that there exis ts a bijectiv e correspondence b etw een all (up to homeomorphism) compact Hausdorff spaces and all (up to isomorphism) complete NCAs. Using the results of F edorc h uk [11] and Ro eper [2 0], w e sho w here that the bijectiv e corresp o ndence established b y Ro ep er can b e extende d to a dua lity b et w een the category Sk eLC of all lo cally compact Hausdorff spaces a nd all sk eletal (in the sense of Mio dusze wski and Rudolf [16]) contin uous maps b etw een them and the category DSk eLC of all complete LCAs and all complete Boolean homomorphisms b et w een them satisfying t wo sim- ple axioms; this is done in Theorem 2.1 1 whic h generalizes the F edorch uk Dualit y Theorem cited ab ov e. F urther, w e r egard the non-full sub category OpLC (resp., OpC ) of t he category Sk eLC (resp., Sk eC ): its ob jects are all lo cally compact (resp., all compact) Hausdorff spaces and its morphisms are all op en maps. W e find the corresp onding sub category DOpLC (resp., DOpC ) of the catego ry DSkeL C (resp., DSke C ) whic h is dually equiv alen t to the category OpLC (resp., OpC ) (see Theorem 2.1 7 and Theorem 2.1 9); as far as w e know, ev en the compact case (i.e. the result ab o ut the catego ry OpC ) is new. The sub categories DSke P erLC and DOpP erLC of the catego ry DSk eLC whic h are dually equiv alen t, respectiv ely , to the categories Sk ePerLC and OpP erLC of all lo cally compact Ha usdorff spaces and all quasi-op en p erfect maps (resp ectiv ely , all op en p erfect maps) b etw een them are fo und as well (see Theorem 2 .15 and Theorem 2.21 ). The ve rsions of all men- tioned ab o v e theorems for the full sub categories of these categories ha ving as o b jects all lo cally compact (resp., compact) connected Hausdorff spaces are formulated (see Theorems 3.2 and 3.3 ) . F ollowing the ideas of F edorc h uk’s pap er [11], w e define fiv e categories EOpC , 2 EOpLC , ESk eLC , ESkeP erLC and EOpP erLC , whic h a re equiv alen t, resp ec- tiv ely , to the categories OpC , OpLC , SkeL C , SkeP erLC and OpP erLC (see Theorems 4.10, 4 .8, 4.4, 4.6, 4.12). The equiv alence b etw een the categories Sk eLC and E SkeLC w as almost established in R o ep er’s pap er [20] (see 4.13 b elo w for more details). The pro of of this equiv alence is a slight mo dification of t he pro of of the analogo us theorem of F edorc h uk [11] concerning the case o f compact Hausdorff spaces. Some further dev elopmen t of the results presen ted here is given in the second part [6] of this pap er. Let us also men tion that in [5] a category dua lly equiv alen t to the category of all lo cally compact Hausdorff spaces and all p erfect maps b etw een them is defined, generalizing in this wa y de V ries Duality Theorem. W e now fix the notations. If C denotes a category , we write X ∈ | C | if X is an ob ject of C , a nd f ∈ C ( X , Y ) if f is a morphism of C with domain X and co domain Y . All lattices are with t o p (= unit) and b ottom (= zero) elemen ts, denoted resp ectiv ely b y 1 and 0. W e do not require t he elemen ts 0 and 1 to be distinct. Let X and Y b e sets. If f : X − → Y is a function then for ev ery subset Z of Y , w e denote by f Z the restriction of f with domain f − 1 ( Z ) and co domain Z , i.e. f Z : f − 1 ( Z ) − → Z . If ( X , τ ) is a to p ological space and M is a subset of X , we denote b y cl ( X,τ ) ( M ) (or simply b y cl( M ) or cl X ( M )) the closure of M in ( X , τ ) and by int ( X,τ ) ( M ) (or briefly by int( M ) or int X ( M )) the in terior of M in ( X , τ ). The Alexandroff compactification of a lo cally compact Hausdorff space X is denoted b y αX . The closed maps and the op en maps b et w een to p ological spaces are assume d to b e con tin uous but are not assumed to b e o n to . Rec all that a map is p erfe ct if it is closed and compact (i.e. p oin t in v erses are compact sets). A con tin uous ma p f : X − → Y is irr e ducible if f ( X ) = Y and if, for eac h pro p er closed subset A of X , f ( A ) 6 = Y . 1 Preliminaries Definition 1.1 An algebraic system B = ( B , 0 , 1 , ∨ , ∧ , ∗ , C ) is called a c ontact algebr a (abbreviated as CA) ([7]) if ( B , 0 , 1 , ∨ , ∧ , ∗ ) is a Bo o lean algebra (where the op eration “complemen t” is denoted b y “ ∗ ”) and C is a binary relation on B , satisfying the follo wing axioms: (C1) If a 6 = 0 then aC a ; (C2) If aC b then a 6 = 0 and b 6 = 0; (C3) aC b implies bC a ; (C4) aC ( b ∨ c ) iff aC b or aC c . Usually , w e shall simply write ( B , C ) for a contact algebra. The relation C is called a c ontact r elation . When B is a complete Bo olean algebra, we will sa y that ( B , C ) is a c omplete c ontact a lgebr a (abbreviated as CCA). F or ev ery tw o subsets M and N of B , w e will write M C N when mC n , for ev ery m ∈ M and ev ery n ∈ N . 3 W e will say that tw o CA’s ( B 1 , C 1 ) and ( B 2 , C 2 ) are CA-isomorphic iff there exists a Bo olean isomorphism ϕ : B 1 − → B 2 suc h that, for eac h a, b ∈ B 1 , aC 1 b iff ϕ ( a ) C 2 ϕ ( b ). Note that in this pap er, by a “Bo o lean isomorphism” w e understand an isomorphism in the category Bo ol . A CA ( B , C ) is called c onne cte d if it satisfies the fo llowing axiom: (CON) If a 6 = 0 , 1 then aC a ∗ . A contact algebra ( B , C ) is called a normal c ontact algebr a (a bbreviated as NCA) ([4, 11]) if it satisfies the follo wing axioms (we will write “ − C ” for “ not C ”): (C5) If a ( − C ) b then a ( − C ) c and b ( − C ) c ∗ for some c ∈ B ; (C6) If a 6 = 1 then there exists b 6 = 0 such that b ( − C ) a . A normal CA is called a c omplete norm a l c ontact algebr a (abbreviated a s CNCA) if it is a CCA. Note that if 0 6 = 1 t hen the axiom (C2) follo ws from the axioms (C6) a nd (C4). F or a n y CA ( B , C ), w e define a binary relation “ ≪ C ” o n B (called n on- tangential i n clusion ) b y “ a ≪ C b ↔ a ( − C ) b ∗ ”. Sometimes w e will write simply “ ≪ ” instead of “ ≪ C ”. The relations C and ≪ a r e in ter- definable. F or example, normal contact alge- bras could b e equiv alen tly defined (and exactly in this w ay they w ere defined (under the name of compingen t Bo olean algebras) by de V ries in [4]) as a pair o f a Bo olean algebra B = ( B , 0 , 1 , ∨ , ∧ , ∗ ) and a binary relation ≪ sub ject t o the following ax- ioms: ( ≪ 1) a ≪ b implies a ≤ b ; ( ≪ 2) 0 ≪ 0; ( ≪ 3) a ≤ b ≪ c ≤ t implies a ≪ t ; ( ≪ 4) a ≪ c and b ≪ c implie s a ∨ b ≪ c ; ( ≪ 5) If a ≪ c then a ≪ b ≪ c for some b ∈ B ; ( ≪ 6) If a 6 = 0 then there exists b 6 = 0 suc h that b ≪ a ; ( ≪ 7) a ≪ b implies b ∗ ≪ a ∗ . Note that if 0 6 = 1 then the axiom ( ≪ 2) f o llo ws from t he a xioms ( ≪ 3), ( ≪ 4) , ( ≪ 6) and ( ≪ 7 ). Ob viously , con tact algebras could b e equiv alen tly defined as a pair of a Bo olean algebra B and a binary relation ≪ sub ject to the axioms ( ≪ 1)-( ≪ 4) and ( ≪ 7). It is easy to see that axiom (C5) (r esp., (C6)) can b e stated equiv alen tly in the form of ( ≪ 5) (resp., ( ≪ 6)). The next notion is a lattice-theoretical counterpart of the corresponding notion from the theory of prox imit y spaces (see [17]): 1.2 Let ( B , C ) b e a CA. Then a non-empt y subset σ of B is called a cluster in ( B , C ) ( see [23]) if the follo wing conditions are satisfied: (K1) If a, b ∈ σ then aC b ; (K2) If a ∨ b ∈ σ t hen a ∈ σ or b ∈ σ ; (K3) If aC b for ev ery b ∈ σ , then a ∈ σ . 4 The set o f all clusters in ( B , C ) will b e denoted b y Clust( B , C ). The set o f all ultrafilters in a Bo olean alg ebra B will be denoted b y Ult( B ). The next three a ssertions can b e prov ed exactly as Lemma 5.6, Theorem 5 .8 and Corollary 5.10 of [17]: F act 1.3 ([23]) I f σ 1 , σ 2 ar e two clusters in a normal c ontact algebr a ( B , C ) and σ 1 ⊆ σ 2 then σ 1 = σ 2 . Theorem 1.4 ([23]) A subset σ o f a normal c ontact algebr a ( B , C ) i s a cluster iff ther e exists an ultr afi l ter u in B such that σ = { a ∈ B | aC b for every b ∈ u } . (1) Mor e over, give n σ an d a 0 ∈ σ , ther e exists an ultr afilter u i n B satisfying (1) which c ontains a 0 . Note that everywher e in this assertion we c an substitute the wor d “ ultr afilter” for “ b asis of a n ultr afilter”. Corollary 1.5 ([23]) L et ( B , C ) b e a normal c ontact algeb r a and u b e an ultr afilter (or a b asis of an ultr afilter) in B . Th en ther e exists a uniq ue cluster σ u in ( B , C ) c ontaining u , and σ u = { a ∈ B | aC b for every b ∈ u } . (2) Definition 1.6 In analogy to the corresp onding definitions in the t heory of pro x- imit y spaces (see, e.g., [1 7]), w e sa y that: (a) a subset ξ of an NCA ( B , C ) is called an end if the following conditions are satisfied: (E1) for an y b, c ∈ ξ there exists a ∈ ξ suc h that a 6 = 0, a ≪ b and a ≪ c ; (E2) if a, b ∈ B and a ≪ b then either a ∗ ∈ ξ or b ∈ ξ ; (b) a subset v of an NCA ( B , C ) is called a r ound filter if it is a filter and for ev ery b ∈ v there exists a ∈ v such that a ≪ b . The next tw o theorems (and their pro of s) a re analogous to the Theorems 6 .7 and 6.11 in [17] (and their pro ofs), resp ectiv ely: Theorem 1.7 L et ( B , C ) b e a normal c ontact alge b r a and ξ b e an end in ( B , C ) . Then ξ is a maxim al r ound filter in ( B , C ) . Theorem 1.8 L et ( B , C ) b e a normal c ontact algebr a an d σ ⊆ B . Then σ ∈ Clust( B , C ) iff d ( σ ) = { b ∈ B | b ∗ 6∈ σ } is an end in ( B , C ) . Corollary 1.9 L et ( B , C ) b e a normal c ontact algebr a, σ ∈ Clust ( B , C ) , a ∈ B and a 6∈ σ . Then ther e exists b ∈ B such that b 6∈ σ and a ≪ b . 5 Pr o of. Put ξ = d ( σ )(= { c ∈ B | c ∗ 6∈ σ } ). Then, b y 1.8 and 1.7, ξ is a round filter in ( B , C ). Since a 6∈ σ , w e obtain that a ∗ ∈ ξ . Hence, there exists b ∗ ∈ ξ suc h that b ∗ ≪ a ∗ . Then b 6∈ σ and a ≪ b . 1.10 Recall that a subset F of a top olog ical space ( X , τ ) is called r e gular close d if F = cl(in t ( F )). Clearly , F is regular closed iff it is a closure of an op en set. F or any top o lo gical space ( X , τ ) , the collection RC ( X, τ ) (w e will often write simply RC ( X )) of a ll regular closed subsets of ( X , τ ) b ecomes a comple te Bo olean algebra ( RC ( X , τ ) , 0 , 1 , ∧ , ∨ , ∗ ) under the fo llo wing op erations: 1 = X , 0 = ∅ , F ∗ = cl( X \ F ) , F ∨ G = F ∪ G, F ∧ G = cl(in t ( F ∩ G )) . The infinite op eratio ns are giv en b y the form ulas W { F γ | γ ∈ Γ } = cl( S { F γ | γ ∈ Γ } )(= cl( S { in t( F γ ) | γ ∈ Γ } )) , and V { F γ | γ ∈ Γ } = cl(in t( T { F γ | γ ∈ Γ } )) . It is easy to see that setting F ρ ( X,τ ) G iff F ∩ G 6 = ∅ , w e define a con ta ct relation o n RC ( X, τ ); it is called a standar d c ontact r elation . So, ( RC ( X, τ ) , ρ ( X,τ ) ) is a CCA (it is called a standar d c ontact algebr a ). W e will often write simply ρ X instead of ρ ( X,τ ) . Note that, for F , G ∈ RC ( X ), F ≪ ρ X G iff F ⊆ int X ( G ). Clearly , if ( X , τ ) is a normal Hausdorff space then the standard con t a ct algebra ( RC ( X, τ ) , ρ ( X,τ ) ) is a complete NCA. F or ev ery top ological space ( X , τ ), w e denote b y R O ( X , τ ) (or simply b y RO ( X )) the set of all regular op en subsets of X (recall that a subset is r e gular op en if its compleme n t is regular closed). F act 1.11 ([2]) L et ( X , τ ) b e a top olo gic al sp ac e. Then the standar d c ontact algebr a ( RC ( X, τ ) , ρ ( X,τ ) ) is c onne cte d iff the sp ac e ( X , τ ) is c onne cte d. Notation 1.12 Let ( X, τ ) b e a t o p ological space and x ∈ X . Then w e set: σ x = { F ∈ R C ( X ) | x ∈ F } and ν x = { F ∈ RC ( X ) | x ∈ in t( F ) } . (3) (Since in o ur notations the p oints of a top o lo gical space are denoted only b y the letters “x,y ,z”, there will b e no confusion with the nota tion σ u in tro duced in 1.5.) F act 1.13 F or any top olo gic al sp ac e ( X , τ ) and ev ery p oint x ∈ X , ν x is a filter in RC ( X ) . I f X is r e gular then σ x is a cluster in the CA ( RC ( X ) , ρ X ) . The next notio n is a lattice-theoretical coun terpart of the Leader’s notio n of lo c al pr oximity ([1 4 ]): Definition 1.14 ([20]) An algebraic system B l = ( B , 0 , 1 , ∨ , ∧ , ∗ , ρ, I B) is called a lo c al c ontact algebr a (abbreviated as LCA) if ( B , 0 , 1 , ∨ , ∧ , ∗ ) is a Bo olean algebra, ρ is a binary relatio n o n B suc h that ( B , ρ ) is a CA, and I B is an ideal (p ossibly non prop er) of B , satisfying the follow ing axioms: 6 (BC1) If a ∈ I B, c ∈ B and a ≪ ρ c then a ≪ ρ b ≪ ρ c for some b ∈ I B (see 1.1 for “ ≪ ρ ”); (BC2) If aρb then there exists an elemen t c of I B suc h that aρ ( c ∧ b ); (BC3) If a 6 = 0 then there exists b ∈ I B \ { 0 } suc h that b ≪ ρ a . Usually , w e shall simply write ( B , ρ, I B) for a lo cal con t a ct algebra. W e will sa y tha t the elemen ts o f I B a re b ounde d and the eleme n ts of B \ I B are unb ounde d . When B is a complete Bo olean algebra, the LCA ( B , ρ, I B) is called a c omplete lo c al c ontact algebr a (a bbreviated by CLCA). W e will say that t w o lo cal con tact algebras ( B , ρ, I B) a nd ( B 1 , ρ 1 , I B 1 ) are LCA-isomorphic iff there exists a Bo olean isomorphism ϕ : B − → B 1 suc h that, for a, b ∈ B , aρb iff ϕ ( a ) ρ 1 ϕ ( b ), and ϕ ( a ) ∈ I B 1 iff a ∈ I B. Remark 1.15 Note that if ( B , ρ, I B) is a lo cal con tact algebra and 1 ∈ I B then ( B , ρ ) is a no r mal con tact algebra. Con v ersely , a n y normal con tact algebra ( B , C ) can b e regarded as a lo cal con ta ct algebra of the form ( B , C , B ). The following lemmas from [23] are lattice-theoretical coun terparts of some theorems from Leader’s pap er [1 4]. Lemma 1.16 ([23]) L et ( B , ρ, I B) b e a lo c al c ontact algebr a. Define a binary r elation “ C ρ ” on B by aC ρ b iff aρb or a, b 6∈ I B . (4) Then “ C ρ ”, c al le d the Alexandroff extension of ρ , is a normal c ontact r elation on B and ( B , C ρ ) is a normal c on tact algebr a. Lemma 1.17 ([23]) L et B l = ( B , ρ, I B) b e a lo c al c ontact algeb r a and let 1 6∈ I B . Then σ B l ∞ = { b ∈ B | b 6∈ I B } is a cluster in ( B , C ρ ) (se e 1.16 for the notation “ C ρ ”). (Sometimes we wil l simp ly write σ ∞ or σ B ∞ inste ad of σ B l ∞ .) Definition 1.18 Let ( B , ρ, I B) b e a lo cal contact algebra. A cluster σ in ( B , C ρ ) (see 1.16) is called b o und e d if σ ∩ I B 6 = ∅ . The set of all b ounded clusters in ( B , C ρ ) will b e denoted b y BClust( B , ρ, I B). An ultrafilter u in B is called a b ounde d ultr afilter if u ∩ I B 6 = ∅ . Notation 1.19 Let ( X , τ ) b e a top ological space. W e will denote b y C R ( X , τ ) the family of all compact regular closed subsets of ( X , τ ). W e will often write C R ( X ) instead of C R ( X , τ ). F act 1.20 L et ( X , τ ) b e a lo c al ly c omp act Hausdorff sp ac e. Then the triple ( RC ( X, τ ) , ρ ( X,τ ) , C R ( X , τ )) (se e 1.10 f o r ρ ( X,τ ) ) is a c om plete lo c al c ontact algebr a ( [ 2 0]). I t is c al le d a standard lo cal contact alg ebra . F or every x ∈ X , σ x is a b ounde d cluster in ( R C ( X ) , C ρ X ) (se e (3) and (4) for the notations) ([23]). 7 1.21 Let ϕ : A − → B b e a n (order-preserving) map b etw een p o sets, A has all meets and ϕ preserv es them. Then, by the Adjoin t F unctor Theorem ( see, e.g., [13]), ϕ has a left adjoint; it will b e denoted by ϕ Λ . Hence ϕ Λ : B − → A is the unique order-preserving map suc h that, for all a ∈ A and all b ∈ B , b ≤ ϕ ( a ) iff ϕ Λ ( b ) ≤ a (i.e. the pair ( ϕ Λ , ϕ ) forms a Galois connection b etw een p osets B and A ). Equiv alen tly , ϕ Λ : B − → A is the unique order-pr eserving map such that the follo wing tw o conditions are f ulfilled: (Λ1) ∀ b ∈ B , ϕ ( ϕ Λ ( b )) ≥ b ; (Λ2) ∀ a ∈ A , ϕ Λ ( ϕ ( a )) ≤ a . It is w ell known that ϕ ◦ ϕ Λ ◦ ϕ = ϕ , ϕ Λ ◦ ϕ ◦ ϕ Λ = ϕ Λ , ϕ Λ preserv es all j oins whic h exist in B (5) and, for all b ∈ B , ϕ Λ ( b ) = ^ { a ∈ A | ϕ ( a ) ≥ b } . (6) F urther, ϕ is a n injection iff ϕ Λ ( ϕ ( a )) = a, ∀ a ∈ A ; (7) ϕ is a surjection iff ϕ ( ϕ Λ ( b )) = b, ∀ b ∈ B . (8) Note that if ϕ (0) = 0 t hen: (a) ϕ Λ (0) = 0 ( use (Λ2)), and (b) ϕ Λ ( b ) 6 = 0, for eve ry b ∈ B \ { 0 } (use (Λ1)). Recall that if ϕ ′ : B − → C is a map b etw een p osets, B has a ll meets and ϕ ′ preserv es them, then ( ϕ ′ ◦ ϕ ) Λ = ϕ Λ ◦ ϕ ′ Λ . Finally , if ψ : A − → B is an (order-preserving) map b et w een p osets, A has all joins and ψ preserv es them, then, by the Adjo in t F unctor Theorem, ψ has a right adjoin t; it will b e denoted b y ψ P ; ψ P : B − → A preserv es all meets whic h exist in B ; setting ϕ = ψ P , w e ha v e that ψ = ϕ Λ . F act 1.22 If A and B ar e Bo ole an algebr as, ϕ : A − → B is a Bo ole an homo m or- phism, A has a l l me ets a n d ϕ pr eserves them , then: (a) ∀ a ∈ A and ∀ b ∈ B , ϕ ( a ) ∧ b = 0 implies a ∧ ϕ Λ ( b ) = 0 ; (b) ∀ a ∈ A and ∀ b ∈ B , ϕ Λ ( ϕ ( a ) ∧ b ) = a ∧ ϕ Λ ( b ) . Pr o of. (a) Let a ∈ A , b ∈ B and ϕ ( a ) ∧ b = 0. Supp ose that a ∧ ϕ Λ ( b ) 6 = 0. Put c = a ∧ ϕ Λ ( b ). If ϕ ( c ) ∧ b = 0 then b ≤ ϕ ( c ∗ ) a nd hence ϕ Λ ( b ) ≤ c ∗ ; therefore c ≤ c ∗ , i.e. c = 0, a con tra diction. Th us ϕ ( c ) ∧ b 6 = 0. This implies that ϕ ( a ) ∧ b 6 = 0, a con tra diction. Therefore, a ∧ ϕ Λ ( b ) = 0. (b) Obviously , ϕ Λ ( ϕ ( a ) ∧ b ) ≤ ϕ Λ ( ϕ ( a )) ∧ ϕ Λ ( b ) ≤ a ∧ ϕ Λ ( b ) (by (Λ2) (see 1 .21)). Hence, w e need only to sho w that ϕ Λ ( ϕ ( a ) ∧ b ) ≥ a ∧ ϕ Λ ( b ). By (6) (see 1.21), w e ha v e to pro v e that a ∧ ϕ Λ ( b ) ≤ V { c ∈ B | ϕ ( c ) ≥ ϕ ( a ) ∧ b } . Let c ∈ B and 8 ϕ ( c ) ≥ ϕ ( a ) ∧ b . W e will sho w that a ∧ ϕ Λ ( b ) ≤ c . Using (a) a nd (Λ1) ( see 1.21), w e obtain that: a ∧ ϕ Λ ( b ) ≤ c ↔ c ∗ ∧ a ∧ ϕ Λ ( b ) = 0 ↔ ϕ ( c ∗ ∧ a ) ∧ b = 0 ↔ ϕ ( c ) ∗ ∧ ϕ ( a ) ∧ b = 0 ↔ ϕ ( a ) ∧ b ≤ ϕ ( c ). Thu s a ∧ ϕ Λ ( b ) ≤ c . Hence (b) is prov ed. F or all undefin ed here notions and notations see [13, 1, 9 , 17, 21]. 2 Some New Dualit y Theo r e ms The next theorem w as pro ved by Ro eper [20]. W e will giv e a sk etch of its pro of; it follo ws the plan o f the pro of presen ted in [23]. The notations and t he facts stated here will be used later on. Theorem 2.1 (P . Ro ep er [2 0]) Ther e e xists a bije ctive c orr esp on denc e b etwe en the class of al l (up to isomorphism) c omplete lo c al c ontact alg ebr as and the class of al l (up to home omorphism) lo c a l ly c omp act Hausdorff sp ac es. Sketch of the Pr o of. (A) Let ( X, τ ) be a lo cally compact Hausdorff space. W e put Ψ t ( X , τ ) = ( RC ( X, τ ) , ρ ( X,τ ) , C R ( X , τ )) (9) (see 1.20 and 1.19 for the notations). (B) Let B l = ( B , ρ, I B) b e a complete lo cal contact algebra. Let C = C ρ b e the Alexandroff extension of ρ ( see 1.16 ). Then, b y 1 .16, ( B , C ) is a complete normal con ta ct algebra. Put X = Clust( B , C ) a nd let T b e the to p ology on X hav ing as a closed base t he family { λ ( B ,C ) ( a ) | a ∈ B } where, f o r ev ery a ∈ B , λ ( B ,C ) ( a ) = { σ ∈ X | a ∈ σ } . (10) Sometimes w e will write simply λ B instead of λ ( B ,C ) . Note that X \ λ B ( a ) = in t( λ B ( a ∗ )) , (11) the family { in t( λ B ( a )) | a ∈ B } is an op en base of ( X , T ) (12) and, for ev ery a ∈ B , λ B ( a ) ∈ R C ( X , T ) . (13) It can b e pro ve d that λ B : ( B , C ) − → ( RC ( X ) , ρ X ) is a CA-isomorphism. (14) F urther, ( X , T ) is a compact Ha usdorff space. (15) (B1) Let 1 ∈ I B. Then C = ρ and I B = B , so that ( B , ρ, I B) = ( B , C , B ) = ( B , C ) is a complete normal con tact alg ebra (see 1.15), and w e put Ψ a ( B , ρ, I B) = Ψ a ( B , C , B ) = Ψ a ( B , C ) = ( X , T ) . (16) 9 (B2) Let 1 6∈ I B. Then, by Lemma 1 .17, the set σ ∞ = { b ∈ B | b 6∈ I B } is a cluster in ( B , C ) and, hence, σ ∞ ∈ X . Let L = X \ { σ ∞ } . Then L = BClust( B , ρ, I B) , i.e. L is the set of all b ounded clusters of ( B , C ρ ) (17) (sometimes w e will write L B l or L B instead of L ); let t he top ology τ (= τ B l ) on L b e the subspace top ology , i.e. τ = T | L . Then ( L, τ ) is a locally compact Hausdorff space. W e put Ψ a ( B , ρ, I B) = ( L, τ ) . (18) Let λ l B l ( a ) = λ ( B ,C ρ ) ( a ) ∩ L, (19) for eac h a ∈ B . W e will write simply λ l B (or ev en λ ( A,ρ, I B) when I B 6 = A ) instead of λ l B l when this does not lead to ambiguit y . One can sho w that: (I) L is a dense subset o f X ; (I I) λ l B is an isomorphism of the Bo olean algebra B onto the Bo olean algebra RC ( L, τ ); (I I I) b ∈ I B iff λ l B ( b ) ∈ C R ( L ); (IV) aρb iff λ l B ( a ) ∩ λ l B ( b ) 6 = ∅ . Hence, X is the Alexandroff (i.e. one-p oin t) compactification of L and λ l B : ( B , ρ, I B) − → ( RC ( L ) , ρ L , C R ( L )) is an LCA-isomorphism. (20) Note also that for ev ery b ∈ B , in t L B ( λ l B ( b )) = L B ∩ in t X ( λ B ( b )) . (21) (C) F o r ev ery CLCA ( B , ρ, I B) and ev ery a ∈ B , set λ g B l ( a ) = λ ( B ,C ρ ) ( a ) ∩ Ψ a ( B , ρ, I B) . (22) W e will write simply λ g B instead of λ g B l when this do es not lead to ambiguit y . Th us, when 1 ∈ I B, w e hav e t hat λ g B = λ B , and if 1 6∈ I B then λ g B = λ l B . Hence, by (14) and (20) , w e get that λ g B : ( B , ρ, I B) − → (Ψ t ◦ Ψ a )( B , ρ, I B) is a n LCA-isomorphism. (23) With the nex t asse rtion we sp ecify ( 1 2): the family { in t Ψ a ( B ,ρ, I B) ( λ g B ( a )) | a ∈ I B } is an op en base of Ψ a ( B , ρ, I B) . (24) (D) Let ( X , τ ) b e a compact Hausdorff space. Then it can b e pro ved that the map t ( X,τ ) : ( X, τ ) − → Ψ a (Ψ t ( X , τ )) , (25) defined by t ( X,τ ) ( x ) = { F ∈ RC ( X, τ ) | x ∈ F } (= σ x ), for a ll x ∈ X , is a homeo- morphism (we will also write simply t X instead of t ( X,τ ) ). 10 Let ( L, τ ) b e a non-compact lo cally compact Hausdorff space. Put B = RC ( L, τ ), I B = C R ( L, τ ) and ρ = ρ L . Then ( B , ρ, I B) = Ψ t ( L, τ ) and 1 6∈ I B (here 1 = L ). It can b e sho wn that the map t ( L,τ ) : ( L, τ ) − → Ψ a (Ψ t ( L, τ )) , (26) defined b y t ( L,τ ) ( x ) = { F ∈ R C ( L, τ ) | x ∈ F } (= σ x ), for all x ∈ L , is a ho meomor- phism. Therefore Ψ a (Ψ t ( L, τ )) is homeomorphic to ( L, τ ) and Ψ t (Ψ a ( B , ρ, I B)) is LCA- isomorphic to ( B , ρ, I B). Corollary 2.2 (De V ries [4]) T her e exists a bije ctive c orr esp on d enc e b etwe en the class of al l (up to isomo rphism) c omplete normal c ontact al g e br as a n d the class of al l (up to home o morphism) c omp act Hausdorff sp a c es. Pr o of. The restriction of the corresp ondence Ψ a , defined in the pro of of Theorem 2.1, to the class of a ll complete normal con tact algebras generates the required bijective corresp ondence (see (B1) in the pro of of 2.1 ). Definition 2.3 ([15]) A con tin uo us map f : X − → Y is called quasi - o p en if for ev ery non-empt y op en subset U of X , in t( f ( U )) 6 = ∅ holds. Ev ery closed irreducible map f : X − → Y is quasi-op en (b ecause, for ev ery non-empt y op en subset U of X , f # ( U )(= { y ∈ Y | f − 1 ( y ) ⊆ U } ) is a non- empty op en subset of Y ([18])). Recall that a f unction f : X − → Y is called skele tal ([16]) if in t ( f − 1 (cl( V ))) ⊆ cl ( f − 1 ( V )) (27) for ev ery op en subset V of Y . As it is noted in [16], a contin uous map f : X − → Y is sk eletal iff f − 1 (F r( V )) is nowhere dense in X , for ev ery op en subset V of Y . Clearly , a function f : X − → Y is sk eletal iff in t( f − 1 (F r( V ))) = ∅ , for ev ery op en subse t V of Y . The next assertion can b e easily pro v ed: Lemma 2.4 A function f : X − → Y is skeletal iff in t (cl( f ( U ))) 6 = ∅ , for ev ery non-empty op en subset U of X . Pr o of. ( ⇒ ) Let U be a non-empt y op en subset of X . Supp ose that in t(cl( f ( U ))) = ∅ . Set V = Y \ cl( f ( U )). Then F r( V ) = Y \ V = cl( f ( U )) and hence U ⊆ f − 1 (F r( V )). Th us in t ( f − 1 (F r( V ))) 6 = ∅ , a contradiction. Therefore, in t(cl( f ( U ) )) 6 = ∅ . ( ⇐ ) Let V b e an op en subset of Y . Su pp ose that U = in t ( f − 1 (F r( V ))) is a non-empt y set. Then ∅ 6 = in t(cl( f ( U ))) ⊆ F r( V ) = cl( V ) \ V , whic h is imp ossible. Hence int( f − 1 (F r( V ))) = ∅ . So, f is a sk eletal map. A top ological space ( X , τ ) is said to b e π - r e gular if f or each non-empt y U ∈ τ there exists a non-empt y V ∈ τ suc h that cl( V ) ⊆ U . The semiregular π -regular spaces are ex actly the we a k ly r e gular spaces of D ¨ un tsch and Winter ([8]). 11 Corollary 2.5 (a) Eve ry quasi-op e n map is skeletal. (b) L et X b e a π -r e gular sp ac e a nd f : X − → Y b e a clos e d map. Then f is quasi-op en iff f is skeletal. Pr o of. (a) It follows from 2.4. (b) Let f b e sk eletal and closed. T ak e an o p en non- empt y subset U of X . Then there exists an op en non-empty subset V o f X suc h that cl( V ) ⊆ U . Using 2.4 , w e obtain that int( f ( U )) ⊇ int( f (cl( V ))) = in t(cl( f ( V ))) 6 = ∅ . Therefore, f is a quasi-op en map. Lemma 2.6 L et f : X − → Y b e a c ontinuous map. Then the fol lowing c onditions ar e e quivalent: (a) f is a skeletal map; (b) F or every F ∈ RC ( X ) , cl( f ( F )) ∈ R C ( Y ) . Pr o of. (a) ⇒ (b) Let f b e a sk eletal map, F ∈ R C ( X ) and F 6 = ∅ . Set U = in t ( F ). Then U 6 = ∅ . Hence, b y 2.4 , V = in t (cl( f ( U ))) 6 = ∅ . W e will show that cl( f ( F )) = cl ( V ) . (28) Note that, b y the contin uit y of f , cl( f ( F )) = cl ( f ( U )). No w suppose tha t f ( U ) 6⊆ cl( V ). Then there exists y ∈ f ( U ) \ cl( V ). Hence there exists an op en neighborho o d O 1 of y in Y suc h that O 1 ∩ V = ∅ . Th us cl( O 1 ) ∩ V = ∅ . There exists x ∈ U suc h that y = f ( x ). Since f is con tin uous, there exists a n op en neigh b orho o d O of x in X suc h that x ∈ O ⊆ U and f ( O ) ⊆ O 1 . Then cl ( f ( O )) ⊆ cl( O 1 ) and t h us cl( f ( O )) ∩ V = ∅ . Since, b y 2.4, ∅ 6 = in t(cl( f ( O ))) ⊆ cl( f ( O )) ∩ int(cl( f ( U ))) = cl ( f ( O )) ∩ V = ∅ , w e obtain a con tradiction. The refore f ( U ) ⊆ cl( V ) and henc e cl( f ( U )) ⊆ cl ( V ). Since the con v erse inclusion is ob vious, (28) is established. Th us, cl( f ( F )) ∈ RC ( Y ). (b) ⇒ (a) Let U b e a non-empt y op en subset of X . Then F = cl( U ) ∈ RC ( X ). Hence cl( f ( F )) ∈ RC ( Y ). Since F 6 = ∅ , we obtain that int(cl( f ( F ))) 6 = ∅ . Now, using the contin uit y o f f , w e g et that in t(cl( f ( U ))) 6 = ∅ . Therefore, b y 2.4, f is a sk eletal map. The next lemm a generalizes the well-kno wn r esult of P o no marev [18] that the regular closed sets are preserv ed b y the closed irreducible maps. Lemma 2.7 L et f : X − → Y b e a clo s e d map an d X b e a π -r e gular sp ac e. Then the fol lowing c on ditions ar e e quivalent: (a) f is a quasi-op en map; (b) F or every F ∈ RC ( X ) , f ( F ) ∈ RC ( Y ) . Pr o of. (a) ⇒ (b) It follows from 2.5(a) and 2.6. (b) ⇒ (a) It follow s from 2.5(b) and 2.6. Note that the π -regularit y of X is used only in the proo f of this implication. 12 Corollary 2.8 If f : X − → Y is a quasi-op en close d map then f ( X ) ∈ RC ( Y ) . Remarks 2.9 In [12], Henriksen a nd Jerison regarded functions f : X − → Y for whic h cl(in t( f − 1 ( F ))) = cl( f − 1 (in t ( F ))) for ev ery F ∈ RC ( Y ) . (29) Clearly , ev ery contin uous sk eletal map f : X − → Y satisfies (29) ([16]). Hence , b y 2.5(a), ev ery quasi-op en map f : X − → Y satisfies (29) ([1 9 ]). F unctions f : X − → Y (not necess arily con t inuous) satisfying condition (27) for ev ery V ∈ RO ( X ) are called HJ-maps in [1 6]. Ob viously , ev ery con tin uous HJ-map f : X − → Y satisfies (29). As it is no ted in [16], the comp osition of t wo con tinuous HJ-maps needs not b e an HJ-map, while the comp osition of t w o con tinuous sk eletal maps is a sk eletal map. It is clear tha t the comp osition of t w o quasi-op en maps is a quasi-op en map. Definition 2.10 Let Sk eLC be the category of all lo cally compact Hausdorff spaces and all contin uous sk eletal maps b et w een them. Let DSk eLC b e the category whose ob jects are all complete lo cal contact algebras a nd whose morphisms ϕ : ( A, ρ, I B) − → ( B , η , I B ′ ) are a ll complete Bo olean homomorphisms ϕ : A − → B satisfying the fo llowing conditions: (L1) ∀ a, b ∈ A , ϕ ( a ) η ϕ ( b ) implies aρb ; (L2) b ∈ I B ′ implies ϕ Λ ( b ) ∈ I B (see 1 .21 for ϕ Λ ). It is easy t o see that in this wa y we hav e define d categories. Let us no t e that (L1) is equiv alent to the follo wing condition: (EL1) ∀ a, b ∈ B , aη b implies ϕ Λ ( a ) ρϕ Λ ( b ). Theorem 2.11 The c ate gories Sk eLC and DSk eLC ar e dual ly e quivalent. Pr o of. W e will define t w o contra v ariant functors Ψ a : DSkeLC − → SkeLC and Ψ t : Sk eLC − → DSk eLC . On the ob jects they coincide with the correspo ndences Ψ a and Ψ t , resp ectiv ely (see (9), (16) and (18 ) for them). W e will define Ψ a and Ψ t on the morphisms of the corresp o nding categories. Let f ∈ Sk eLC ( ( X, τ ) , ( Y , τ ′ )). D efine Ψ t ( f ) : Ψ t ( Y , τ ′ ) − → Ψ t ( X , τ ) b y Ψ t ( f ) ( F ) = cl( f − 1 (in t( F ))) , ∀ F ∈ Ψ t ( Y , τ ′ ) . (30) Then, by 2.9, Ψ t ( f ) ( F ) = cl(int( f − 1 ( F ))) , ∀ F ∈ Ψ t ( Y , τ ′ ) . (31) Put ϕ = Ψ t ( f ) . W e will first sho w that ϕ is a complete Bo olean homomor phism. Let Γ b e a set and { F γ | γ ∈ Γ } ⊆ RC ( Y ). Put F = cl( S { F γ | γ ∈ Γ } ). (Note that F = cl( S { in t ( F γ ) | γ ∈ Γ } ).) Then F ∈ R C ( Y ) and W { F γ | γ ∈ Γ } = F . Since ϕ is an order-preserving map, w e get that ϕ ( F ) ≥ W { ϕ ( F γ ) | γ ∈ Γ } . W e will now pro ve the con v erse inequalit y . W e ha ve that ϕ ( F ) = cl( f − 1 (in t( F ))). Let x ∈ f − 1 (in t ( F )). Then f ( x ) ∈ int( F ). Hence, there exist op en neighborho o ds O and O ′ of f ( x ) in Y 13 suc h that cl( O ′ ) ⊆ O ⊆ F . Since f is contin uous, there exists an op en neighborho o d U of x in X suc h that f ( U ) ⊆ O ′ . Supp ose that there exists an op en neigh b orho o d V of x in X suc h that, fo r eve ry γ ∈ Γ, V ∩ cl(int( f − 1 ( F γ ))) = ∅ . Obviously , we can supp ose tha t V ⊆ U . Since f is contin uous and ske letal, we get, using 2.9 and (29), that V ∩ f − 1 (in t( F γ )) = ∅ , for ev ery γ ∈ Γ. Th us, f ( V ) ∩ S { in t( F γ ) | γ ∈ Γ } = ∅ . Put W = S { in t( F γ ) | γ ∈ Γ } . Then cl( f ( V )) ∩ W = ∅ and cl ( f ( V )) ⊆ cl( f ( U ) ⊆ cl( O ′ ) ⊆ O ⊆ F = cl( W ). Th us cl( f ( V )) ⊆ cl( W ) \ W = F r( W ). Since f is sk eletal, 2.4 implies that in t (cl( f ( V ))) 6 = ∅ a nd this leads to a contradiction. Therefore, x ∈ cl( S { cl(in t( f − 1 ( F γ ))) | γ ∈ Γ } ). W e ha ve pro ved that ϕ ( F ) ⊆ W { ϕ ( F γ ) | γ ∈ Γ } . So, ϕ ( W { F γ | γ ∈ Γ } ) = W { ϕ ( F γ ) | γ ∈ Γ } . Let F ∈ R C ( Y ). Then, b y (3 0 ) and (3 1), ( ϕ ( F )) ∗ = (cl( f − 1 (in t( F )))) ∗ = (cl( f − 1 ( Y \ F ∗ ))) ∗ = (cl( X \ f − 1 ( F ∗ ))) ∗ = cl( X \ cl ( X \ f − 1 ( F ∗ ))) = cl(int( f − 1 ( F ∗ ))). So, ϕ ( F ∗ ) = ( ϕ ( F )) ∗ . Since, ob viously , ϕ preserv es the zero and the unit elemen ts, ϕ is a complete Bo olean homomorphism. F urther, usin g 2 .6 , w e can define a map ψ : Ψ t ( X , τ ) − → Ψ t ( Y , τ ′ ) b y ψ ( G ) = cl( f ( G )) , for ev ery G ∈ Ψ t ( X , τ ) . (32) Ob viously , ψ is an order-preserving map. Since f is a con tinuous map, we ha ve that for every F ∈ R C ( Y ), ψ ( ϕ ( F )) = cl( f (cl( f − 1 (in t( F ))))) = cl( f ( f − 1 (in t ( F )))) ⊆ cl(in t ( F )) = F , and, similarly , for ev ery G ∈ RC ( X ), ϕ ( ψ ( G )) = ϕ (cl ( f ( G ))) = cl(in t ( f − 1 (cl( f ( G ) )))) ⊇ cl(int( G )) = G . Hence ψ is a left adjoin t to ϕ (see 1.21), i.e. ψ = ϕ Λ . (33) W e ha v e to show that ϕ satisfies conditions (L1) and (L 2 ). Using (33), w e obtain immediately that (EL1) (a nd henc e (L 1 )) and ( L 2) are f ulfilled. Hence, Ψ t ( f ) is a morphism of the category D SkeLC . It is ob vious that Ψ t ( id ) = id . Let us sho w that Ψ t ( g ◦ f ) = Ψ t ( f ) ◦ Ψ t ( g ). Indeed, using con tinuit y of f and g , 2.9 and (29), w e obtain that (Ψ t ( f ) ◦ Ψ t ( g ))( F ) = cl(in t ( f − 1 (cl( g − 1 (in t( F )))))) ⊇ cl (in t(cl( f − 1 ( g − 1 (in t ( F )))))) = Ψ t ( g ◦ f )( F ) and also (Ψ t ( f ) ◦ Ψ t ( g ))( F ) = cl( f − 1 (in t( Ψ t ( g )( F )) ) ) ⊆ cl( int( f − 1 (cl(in t ( g − 1 ( F )))))) ⊆ cl(in t ( f − 1 ( g − 1 ( F )))) = Ψ t ( g ◦ f )( F ). Thus , Ψ t : Sk eLC − → DSk eLC is a contra v arian t functor. Let ϕ ∈ DSkeLC (( A, ρ, I B) , ( B , η , I B ′ )). Since ϕ : A − → B is a complete Bo olean homomorphism, ϕ has a left adjoin t ϕ Λ : B − → A (see 1.2 1). Set C = C ρ and C ′ = C η (see 1.16 for the notations). W e will write “ ≪ ” for “ ≪ C ” and “ ≪ ′ ” for “ ≪ C ′ ”. Define now Ψ a ( ϕ ) : Ψ a ( B , η , I B ′ ) − → Ψ a ( A, ρ, I B) (34) b y the form ula Ψ a ( ϕ )( σ u ) = σ ϕ − 1 ( u ) , (35) 14 where u ∈ Ult( B ), σ u is a cluster in ( B , C ′ ), σ u ∩ I B ′ 6 = ∅ and σ ϕ − 1 ( u ) is a cluster in ( A, C ) (see (2) and 1.5 for σ u and σ ϕ − 1 ( u ) , and note that, by 1.4 , an y cluster σ in ( B , C ′ ) can b e written in t he form σ u for some u ∈ Ult( B )). W e hav e to show that Ψ a ( ϕ ) is w ell defined. Set f = Ψ a ( ϕ ), X = Ψ a ( A, ρ, I B) and Y = Ψ a ( B , η , I B ′ ). Then X is the set of all b ounded clusters of ( A, C ) and Y is the set of all b ounded clusters of ( B , C ′ ) (see 1.1 8, (16) and (17)). Let us start with the fo llo wing observ ation: if u ∈ Ult( B ) then ϕ − 1 ( u ) ∈ Ult( A ) and ϕ Λ ( u ) is a basis of ϕ − 1 ( u ) . (36) So, let u ∈ Ult( B ). Then, ob viously , ϕ − 1 ( u ) ∈ Ult( A ). Let us show that ϕ Λ ( u ) ⊆ ϕ − 1 ( u ). Let b ∈ u . Then, b y (Λ1), ϕ ( ϕ Λ ( b )) ≥ b . Hence ϕ ( ϕ Λ ( b ) ∈ u , i.e. ϕ Λ ( b ) ∈ ϕ − 1 ( u ). Therefore, ϕ Λ ( u ) ⊆ ϕ − 1 ( u ). F urther, suppose that there exists a ∈ ϕ − 1 ( u ) suc h that ϕ Λ ( b ) 6≤ a for all b ∈ u . Then ϕ Λ ( b ) ∧ a ∗ 6 = 0 for ev ery b ∈ u . Hence, b y 1.22(a), b ∧ ϕ ( a ∗ ) 6 = 0 for ev ery b ∈ u . Since u ∈ Ult( B ), we o btain that ϕ ( a ∗ ) ∈ u . Th us b oth ϕ ( a ) and ϕ ( a ) ∗ are elemen ts of u , a con tradiction. Therefore, ϕ Λ ( u ) is a basis of the ultra filter ϕ − 1 ( u ). Ob viously , (36) implies t ha t ∀ u ∈ Ult( B ) , σ ϕ − 1 ( u ) = σ ϕ Λ ( u ) , (37) where σ ϕ − 1 ( u ) and σ ϕ Λ ( u ) are clusters in ( A, C ) (see 1.5 f o r the notat io ns). Let σ b e a cluste r in ( B , C ′ ). Then the following ho lds: if σ ∩ I B ′ 6 = ∅ then there exists b ∈ I B ′ suc h that b ∗ 6∈ σ. (38) Indeed, let b 0 ∈ σ ∩ I B ′ . Since b 0 ≪ η 1, (BC1) implies that there exists b ∈ I B ′ suc h that b 0 ≪ η b . Then b 0 ( − η ) b ∗ and since b 0 ∈ I B ′ , w e obtain tha t b 0 ( − C ′ ) b ∗ . Thu s b ∗ 6∈ σ . Let us now sho w that if u ∈ Ult( B ) and σ u ∩ I B ′ 6 = ∅ then u ∩ I B ′ 6 = ∅ (39) (here, of course, σ u is a cluster in ( B , C ′ )). Indeed, b y (38), there exists a ∈ I B ′ suc h that a ∗ 6∈ σ u . Hence a ∈ u ∩ I B ′ . So, (39) is prov ed. Let u, v ∈ Ult( B ), σ u = σ v and σ = σ u (= σ v ) b e b ounded. W e will pro v e that σ ϕ − 1 ( u ) = σ ϕ − 1 ( v ) . Indeed, by (3 9), there exists c ∈ u ∩ I B ′ . Let a ∈ u and b ∈ v . Then a ∧ c ∈ u ∩ I B ′ and ( a ∧ c ) C ′ b . Thus ( a ∧ c ) η b . Hence, by (EL1), ϕ Λ ( a ∧ c ) ρϕ Λ ( b ). Therefore, ϕ Λ ( a ∧ c ) C ϕ Λ ( b ). Th us ϕ Λ ( a ) C ϕ Λ ( b ). Since this is true for ev ery a ∈ u and ev ery b ∈ v , w e obtain, using (36) and (2), that ϕ Λ ( u ) ⊆ σ ϕ Λ ( v ) . Then, b y 1.5 and (36) , σ ϕ Λ ( u ) = σ ϕ Λ ( v ) . Using (37), we get that σ ϕ − 1 ( u ) = σ ϕ − 1 ( v ) . No w, using (37), w e obtain that if σ ∈ Y and b ∈ σ then ϕ Λ ( b ) ∈ f ( σ ) . (40) Indeed, b y 1 .4 , there exists u ∈ Ult( B ) suc h that b ∈ u ⊆ σ , and hence σ = σ u . Th us, b y (37), f ( σ ) = σ ϕ Λ ( u ) . Therefore ϕ Λ ( b ) ∈ f ( σ ). So, (40) is pro ve d. 15 Let us show that for ev ery σ ∈ Clust( B , C ′ ), σ ∩ I B ′ 6 = ∅ implies that f ( σ ) ∩ I B 6 = ∅ . (41) Indeed, let σ ∈ Clust( B , C ′ ) and b ∈ σ ∩ I B ′ . Then, by (40), ϕ Λ ( b ) ∈ f ( σ ). Since, b y (L2), ϕ Λ ( b ) ∈ I B, w e obta in that f ( σ ) ∩ I B 6 = ∅ . So, the function f is w ell defined on Y and f ( Y ) ⊆ X . W e hav e to show that f is con tin uo us and sk eletal. Note first that, using (11) and (21) , w e get readily that for ev ery a ∈ A , X \ λ g A ( a ) = int X ( λ g A ( a ∗ )) . (42) F urther, usin g (2 3) and 1 .10, one can easily sho w that for all a, b ∈ A , a ≪ ρ b implies t ha t λ g A ( a ) ⊆ int X ( λ g A ( b )) . (43) Note also that if σ is a cluster in ( B , C ′ ) then b ∗ 1 , b ∗ 2 6∈ σ implies that b 1 ∧ b 2 ∈ σ and ( b 1 ∧ b 2 ) ∗ 6∈ σ. (44) Indeed, if b ∗ 1 , b ∗ 2 6∈ σ then, by (K2 ) , b ∗ 1 ∨ b ∗ 2 6∈ σ , i.e. ( b 1 ∧ b 2 ) ∗ 6∈ σ ; hence b 1 ∧ b 2 ∈ σ . Let us now pro v e that for every b ∈ I B ′ , f ( λ g B ( b )) = λ g A ( ϕ Λ ( b )) (45) (note that b ∈ I B ′ implies that λ B ( b ) ⊆ Y and ϕ Λ ( b ) ∈ I B (by (L2)); th us w e hav e also that λ A ( ϕ Λ ( b )) ⊆ X ; hence (45) can b e written a s f ( λ B ( b )) = λ A ( ϕ Λ ( b ))). Since ϕ (0) = 0, we ha v e, b y 1.2 1 , that ϕ Λ (0) = 0 and ϕ Λ ( b ) 6 = 0 for any b 6 = 0. Hence, (45) is true for b = 0. Let b ∈ I B ′ \ { 0 } and σ ∈ f ( λ B ( b )). Then t here exists σ ′ ∈ λ B ( b ) suc h that f ( σ ′ ) = σ . Hence b ∈ σ ′ and th us, by ( 40), ϕ Λ ( b ) ∈ f ( σ ′ ) = σ . Therefore we g et that σ ∈ λ A ( ϕ Λ ( b )). So, f ( λ B ( b )) ⊆ λ A ( ϕ Λ ( b )). C on vers ely , let b ∈ I B ′ \ { 0 } and σ ∈ λ A ( ϕ Λ ( b )), i.e. ϕ Λ ( b ) ∈ σ . Then, b y 1.4, there exists u ∈ Ult( A ) suc h tha t ϕ Λ ( b ) ∈ u ⊆ σ , and hence, by 1.5, σ = σ u . Let us sho w tha t ϕ ( u ) ∪ { b } has the finite in tersection prop erty . Since ϕ ( u ) is closed under finite meets, it is enough to pro v e t hat b ∧ ϕ ( a ) 6 = 0 , ∀ a ∈ u . Indeed, supp ose that t here exists a 0 ∈ u suc h that b ∧ ϕ ( a 0 ) = 0 . Then, b y 1.22(a ) , w e will hav e that ϕ Λ ( b ) ∧ a 0 = 0. This is, ho w ev er, imp ossible, since ϕ Λ ( b ) ∈ u . So, there exists an ultrafilter v in B suc h that v ⊇ ϕ ( u ) ∪ { b } . Set σ ′ = σ v . Then σ ′ is a cluster in ( B , C ′ ) (see 1 .4) and since v ⊆ σ ′ , w e hav e tha t b ∈ σ ′ . Hence σ ′ ∈ λ B ( b ). F urther, f ( σ ′ ) = σ . Indeed, sinc e ϕ ( u ) ⊆ v , w e hav e that u ⊆ ϕ − 1 ( v ); thus u = ϕ − 1 ( v ) and hence σ = σ u = σ ϕ − 1 ( v ) = f ( σ v ) = f ( σ ′ ). Therefore σ = f ( σ ′ ) ∈ f ( λ B ( b )). So , (45) is pro v ed. W e are ready to sho w that f is a con tin uous function. Let σ ∈ Y , σ ′ = f ( σ ), a ∈ A and σ ′ ∈ int X ( λ g A ( a )) (we use (24 )). Then, b y (4 2), a ∗ 6∈ σ ′ . By 1.9, there exists a 1 ∈ A suc h that a ∗ ≪ a ∗ 1 and a ∗ 1 6∈ σ ′ . 16 Then a 1 ∈ v , for ev ery v ∈ Ult( A ) suc h that v ⊆ σ ′ . Th us, using (36), w e obta in that for ev ery u ∈ Ult( B ) suc h that u ⊆ σ , there exists b u ∈ u with ϕ Λ ( b u ) ≤ a 1 . Set b = W { b u | u ∈ Ult( B ) , u ⊆ σ } . Then, by 1.21, ϕ Λ ( b ) = W { ϕ Λ ( b u ) | u ∈ Ult( B ) , u ⊆ σ } . Henc e ϕ Λ ( b ) ≤ a 1 . Supp ose that b ∗ ∈ σ . Then 1.4 implies that there exists u ∈ Ult( B ) such that b ∗ ∈ u ⊆ σ . Since b ∈ u (b ecause b u ∈ u and b u ≤ b ), w e obtain a contradiction. Hence b ∗ 6∈ σ . Since σ is a b ounded cluster, (38) implies that there exists c ∈ I B ′ suc h that c ∗ 6∈ σ . Set d = b ∧ c . Then d ∈ I B ′ and d ∗ 6∈ σ (b y (44)). No w, using ( L 2 ), (43 ) and (45), we o btain that f (int Y ( λ g B ( d ))) ⊆ f ( λ g B ( d )) = λ g A ( ϕ Λ ( d )) ⊆ λ g A ( ϕ Λ ( b )) ⊆ λ g A ( a 1 ) ⊆ in t X ( λ g A ( a )). Since σ ∈ in t Y ( λ g B ( d )), w e g et that f : Y − → X is a con tin uous f unction. (46) W e will no w sho w tha t f is a sk eletal map. Since f is con tin uo us, it is enough to prov e, b y 2.4, t ha t in t X ( f ( cl( U ))) 6 = ∅ for ev ery non-empt y op en subset U of Y . Hence, b y (2 4 ) and (23), w e hav e to sho w t hat int X ( f ( λ g B ( b )) 6 = ∅ , for ev ery b ∈ I B ′ \ { 0 } . Supp ose that there exists b ∈ I B ′ \ { 0 } suc h tha t in t X ( f ( λ g B ( b ))) = ∅ . Then X \ f ( λ g B ( b )) is dense in X . Using (45), we obtain that X \ λ g A ( ϕ Λ ( b )) is dense in X . Th us, b y (4 2), in t( λ g A (( ϕ Λ ( b )) ∗ )) is dense in X . Now , (23 ) implies that λ g A (( ϕ Λ ( b )) ∗ ) = X . T herefore, b y (2 3), ( ϕ Λ ( b )) ∗ = 1. Then ϕ Λ ( b ) = 0 and hence b = 0 ( by 1 .21), a con tradiction. Hence, f : Y − → X is a sk eletal map. (47) So, w e hav e prov ed that Ψ a ( ϕ ) ∈ Sk eLC (Ψ a ( B , η , I B ′ ) , Ψ a ( A, ρ, I B)). Th us Ψ a is w ell defined on the morphisms of the categor y DSk eLC . It is easy to see that Ψ a preserv es the iden tity maps and that Ψ a ( ϕ 1 ◦ ϕ 2 ) = Ψ a ( ϕ 2 ) ◦ Ψ a ( ϕ 1 ). Th us, Ψ a : DSk eLC − → Sk eLC is a contra v arian t functor. W e will pro v e that Ψ a ◦ Ψ t ∼ = I d Ske LC (where “ ∼ = ′′ means “naturally isomor- phic” and I d is the iden tity functor). W e will sho w t hat t : I d Ske LC − → Ψ a ◦ Ψ t , (48) defined by t ( X, τ ) = t ( X,τ ) , ∀ ( X , τ ) ∈ | Sk eLC | , (49) is the required natural isomorphism (see (2 5 ) and (2 6) for the definition of t ( X,τ ) ). Let f ∈ Sk eLC (( X , τ ) , ( Y , τ ′ )) and ˆ f = Ψ a (Ψ t ( f ) ) . W e hav e to sho w that ˆ f ◦ t X = t Y ◦ f . Let x ∈ X . Then ˆ f ( t X ( x )) = ˆ f ( σ x ) a nd ( t Y ◦ f )( x ) = σ f ( x ) . W e will pro v e that ˆ f ( σ x ) = σ f ( x ) . (50) 17 Note first that if u ∈ Ult( RC ( X )) , x ∈ X and u ⊃ ν x then u ⊂ σ x (51) (see (3) for ν x ). Indeed, let F ∈ u and supp ose that x 6∈ F . Then x ∈ X \ F = in t( F ∗ ) and hence F ∗ ∈ ν x . Th us F ∗ ∈ u , a con tradiction. So, u ⊂ σ x . Set Ψ t ( f ) = ϕ . Let x ∈ X . Since ν x is a filter in RC ( X ) (see 1.13), there exists u ∈ Ult( RC ( X )) suc h that ν x ⊆ u . Then, by (51) and 1.5, σ x = σ u . Hence ˆ f ( σ x ) = σ ϕ − 1 ( u ) . W e will no w sho w that ν f ( x ) ⊆ ϕ − 1 ( u ). Indeed, let G ∈ RC ( Y ) and f ( x ) ∈ in t Y ( G ). Then, b y the contin uit y of f , x ∈ f − 1 (in t Y ( G )) ⊆ in t X ( ϕ ( G )) ( see (31)). Th us ϕ ( G ) ∈ ν x ⊆ u . Hence G ∈ ϕ − 1 ( u ). Therefore ν f ( x ) ⊆ ϕ − 1 ( u ). Then, by (51) and 1.5, σ f ( x ) = σ ϕ − 1 ( u ) = ˆ f ( σ x ). So, w e hav e pro v ed that ˆ f ( t X ( x )) = t Y ( f ( x )), for ev ery x ∈ X . Hence, t is a natura l isomorphism. Finally , we will prov e that Ψ t ◦ Ψ a ∼ = I d DSk eLC . W e will sho w t hat λ : I d DSk eLC − → Ψ t ◦ Ψ a , whe re λ ( A, ρ, I B) = λ g A , ∀ ( A, ρ, I B) ∈ | DSkeL C | (52) (see (22) for λ g A ), is the require d natural isomorphism. Let ( A, ρ, I B) ∈ | DSkeL C | . Using (23), it is easy to see that λ g A : ( A, ρ, I B) − → Ψ t (Ψ a ( A, ρ, I B)) is an DSk eLC -isomorphism. (53) Let ϕ ∈ DSk eLC (( A, ρ, I B) , ( B , η , I B ′ )) and ˆ ϕ = Ψ t (Ψ a ( ϕ )). W e ha v e to pro v e that λ g B ◦ ϕ = ˆ ϕ ◦ λ g A . Set f = Ψ a ( ϕ ). Let a ∈ A \ { 0 } . P ut F = λ g A ( a ) and G = λ g B ( ϕ ( a )). W e ha v e to sho w that ˆ ϕ ( F ) = G , i.e. that G = cl( f − 1 (in t ( F )))(= cl(int( f − 1 ( F )))). Let σ ∈ G . Then ϕ ( a ) ∈ σ and σ ∩ I B ′ 6 = ∅ . Th us, by (40), ϕ Λ ( ϕ ( a )) ∈ f ( σ ). Using (Λ2 ), w e obtain that a ∈ f ( σ ). Therefore f ( σ ) ∈ λ g A ( a ) = F . So, σ ∈ f − 1 ( F ). W e hav e shown that G ⊆ f − 1 ( F ). Then in t ( G ) ⊆ cl(in t ( f − 1 ( F ))) and th us G ⊆ ˆ ϕ ( F ). Con v ersely , let σ ∈ cl( f − 1 (in t ( F ))). Supp ose that ϕ ( a ) 6∈ σ . Then σ ∈ in t( λ g B ( ϕ ( a ∗ ))) (see (42)). Hence in t ( λ g B ( ϕ ( a ∗ ))) ∩ f − 1 (in t( F )) 6 = ∅ . Thus there exists σ ′ ∈ in t ( λ g B ( ϕ ( a ∗ ))) suc h that f ( σ ′ ) ∈ int( F ). Then, by (42) , a ∗ 6∈ f ( σ ′ ). Since ϕ ( a ∗ ) ∈ σ ′ , (40) and (Λ2 ) imply that a ∗ ∈ f ( σ ′ ), a con tra diction. Hence ϕ ( a ) ∈ σ , i.e. σ ∈ G . So, ˆ ϕ ( F ) ⊆ G . Therefore, ˆ ϕ ( F ) = G . This show s that λ g B ◦ ϕ = ˆ ϕ ◦ λ g A . (54) Hence, λ is a natural isomorphism. W e ha v e prov ed that Sk eLC and DSkeLC are dually equiv alen t categories. Definition 2.12 (F edorc huk [1 1]) W e will denote by SkeC the category o f all com- pact Hausdorff spaces a nd all quasi-op en maps b et w een them. Let DSk eC b e the categor y whose ob jects are all complete no r ma l con tact algebras and whose morphisms ϕ : ( A, C ) − → ( B , C ′ ) are all complete Bo o lean homomorphisms ϕ : A − → B satisfying the fo llowing condition: 18 (F1) F or all a, b ∈ A , ϕ ( a ) C ′ ϕ ( b ) implies aC b . It is easy t o see that in this wa y we hav e define d categories. Theorem 2.13 (F edorc huk [11]) The c ate gories Sk eC and DSk eC ar e dual ly e qui- valent. Pr o of. It follows from Theorem 2.11, 1.15 and 2.5 (b). W e will no w obtain o ne more generalization o f Theorem 2.1 3. Definition 2.14 Let Sk ePerLC b e the category of all lo cally compact Ha usdorff spaces and all sk eletal p erfect maps b etw een them. Note that, by 2.5(b), the mor- phisms of the category Sk ePerLC are precisely the quasi-o p en p erfect maps. Let DSkeP erLC b e the category whose ob jects are all complete lo cal con- tact algebras (see 1.14) and whose morphisms are all DSk eLC - morphisms ϕ : ( A, ρ, I B) − → ( B , η , I B ′ ) satisfying the fo llo wing condition: (L3) a ∈ I B implies ϕ ( a ) ∈ I B ′ . It is easy to se e that in this w ay w e hav e defined categories. Ob viously , Sk ePerLC (resp., DSk eP erLC ) is a (non-full) sub category of the category Sk eLC (resp., DSkeLC ). Theorem 2.15 The c ate gories SkeP erLC an d DSk ePerLC ar e dual ly e quivalent. Pr o of. W e will sho w that the restrictions Ψ a p : DSk eP er LC − → SkeP erLC and Ψ t p : Sk eP er LC − → DSk ePerLC of the contra v ariant functors Ψ a and Ψ t defined in the proo f of Theorem 2.11 a r e the desired duality functors. Let f ∈ Sk eP er LC (( X, τ ) , ( Y , τ ′ )). Since f is a perfect map, we obtain that ϕ = Ψ t p ( f ) satisfies condition (L3) (using [9, Theorem 3.7.2]). Hence, ϕ is w ell defined. Therefore t he con tra v arian t functor Ψ t p : Ske P erLC − → DSk ePe rLC is w ell defined. Let ϕ ∈ DSk ePerLC (( A, ρ, I B) , ( B , η , I B ′ )) and set f = Ψ a p ( ϕ ), i.e. f : Ψ a p ( B , η , I B ′ ) − → Ψ a p ( A, ρ, I B). Put C = C ρ and C ′ = C η (see 1.16 for the nota- tions). Then, b y 1 .1 6, ( A, C ) and ( B , C ′ ) are CNCA’s. Denote by ϕ c the map ϕ regarded as a function of ( A, C ) to ( B , C ′ ). W e will show that ϕ c satisfies condition (F1) in 2.12. F or v erifying condition (F1 ) , let a, b ∈ A and let ϕ c ( a ) C ′ ϕ c ( b ). Then either ϕ c ( a ) η ϕ c ( b ) or ϕ c ( a ) , ϕ c ( b ) 6∈ I B ′ . If ϕ c ( a ) η ϕ c ( b ) then, by (L1), aρb ; hence aC b . If ϕ c ( a ) , ϕ c ( b ) 6∈ I B ′ then, b y (L3), a, b 6∈ I B. Hence aC b . So, (F1) is v erified. Therefore, ϕ c : ( A, C ρ ) − → ( B , C η ) satisfies condition (F1). (55) Set X = Ψ a ( A, C , A ) and Y = Ψ a ( B , C ′ , B ) (see (16)) . Then X and Y are compact Hausdorff space s. Let f c = Ψ a ( ϕ c ), i.e. f c : Y − → X is defined b y f c ( σ u ) = σ ϕ − 1 c ( u ) , for eve ry u ∈ Ult( B ) . (56) 19 Then, by (46) , (47) a nd 2.5(b), we obtain that f c : Y − → X is a quasi-open map. (57) W e will regard three cases no w. (a) Let 1 A 6∈ I B and 1 B 6∈ I B ′ . Then Ψ a p ( B , η , I B ′ ) = L B = Y \ { σ B ∞ } and Ψ a p ( A, ρ, I B)) = L A = X \ { σ A ∞ } (see 1.17 a nd (17)). W e will show that f − 1 c ( σ A ∞ ) = { σ B ∞ } (see 1.17 fo r the notations). W e first pr ov e that f c ( σ B ∞ ) = σ A ∞ . Let u ∈ Ult( B ) b e suc h that u ⊂ σ B ∞ and σ B ∞ = σ u (see 1.4) . Then f c ( σ B ∞ ) = σ ϕ − 1 c ( u ) . W e will sho w tha t ϕ − 1 c ( u ) ⊂ σ A ∞ . Inde ed, let a ∈ ϕ − 1 c ( u ). Then ϕ c ( a ) ∈ u ⊂ B \ I B ′ . Hence ϕ c ( a ) 6∈ I B ′ . Th us, by (L3 ) , a 6∈ I B. So, ϕ − 1 c ( u ) ⊂ A \ I B = σ A ∞ (see 1.17). Then, by 1.17 and 1.5, σ A ∞ = σ ϕ − 1 c ( u ) . The refore, f c ( σ B ∞ ) = σ A ∞ . Since L A and L B consist of b ounded clusters (see (17)), (4 1) implies that f c ( L B ) ⊆ L A . Therefore, f − 1 c ( σ A ∞ ) = { σ B ∞ } . This show s that f − 1 c ( L A ) = L B . Since f c is a p erfect map, w e obtain ( by [9 , Prop osition 3.7.4]) that ( f c ) L A : L B − → L A is a p erfect map. (58) Ob viously , f is the restriction of f c to L B . Hence f = ( f c ) L A , i.e. f is a p erfect map. Since f is a sk eletal map (by (47)), 2.5 (b) implies that f is a quasi-op en p erfect ma p. (59) (b) Let 1 A 6∈ I B and 1 B ∈ I B ′ . The n C ′ = η , Ψ a p ( A, ρ, I B) = X \ { σ A ∞ } = L A and Ψ a p ( B , η , I B ′ ) = Y . Th us (41) implies that f c ( Y ) ⊂ L A . Therefore, the restriction f : Y − → L A of f c is a p erfect map. Since f is sk eletal (b y (47)), we obtain, using 2.5(b), that f is quasi-op en. Therefore, f : Ψ a p ( B , η , I B ′ ) − → Ψ a p ( A, ρ, I B) (60) is a quasi-op en p erfect map. (c) L et 1 A ∈ I B. Then, b y (L3), 1 B ∈ I B ′ . Hence C = ρ , C ′ = η , Ψ a p ( B , η , I B ′ ) = Y , Ψ a p ( A, ρ, I B) = X . Th us f = f c . Hence, b y ( 46), (47) and 2.5(b), f : Y − → X is a quasi-op en p erfect map. W e hav e regarded all p o ssible cases. Therefore, Ψ a p is w ell defin ed o n the ob jects and morphisms of the catego ry DSk eP erLC . Note that, using (23), we obta in that λ g B is a DSkeP erLC -isomorphism. The rest follows f rom Theorem 2 .1 1. Definition 2.16 Let OpLC be the catego ry of all lo cally compact Hausdorff spaces and all o p en maps betw een them. Let DOpLC b e the category whose ob jects are all complete lo cal contact alge- bras and whose morphisms are all DSk eLC -mor phisms ϕ : ( A, ρ, I B) − → ( B , η , I B ′ ) satisfying the follo wing condition: 20 (LO) ∀ a ∈ A and ∀ b ∈ I B ′ , ϕ Λ ( b ) ρa implies bη ϕ ( a ). It is easy to se e that in this w ay w e hav e defined categories. Ob viously , DOpLC ( r esp., OpLC ) is a (non- full) sub categor y of the category DSk eLC (resp., Sk eLC ). Theorem 2.17 The c ate gories OpLC and DOpLC ar e dual ly e q uivalent. Pr o of. W e will sho w that the restrictions Ψ a o : DOpLC − → OpLC and Ψ t o : OpLC − → DOpLC of the contra v ariant functors Ψ a and Ψ t defined in the pro of of Theorem 2 .11 are the des ired dualit y functors. Let f ∈ OpLC (( X , τ ) , ( Y , τ ′ )). Set ϕ = Ψ t o ( f ) . T hen, since f is an op en map, [9, 1.4.C] implies t ha t for ev ery F ∈ RC ( Y ), f − 1 ( F ) = f − 1 (cl(in t( F ))) = cl( f − 1 (in t ( F ))) = ϕ ( F ) (see (30)). Hence, Ψ t o ( f ) : Ψ t o ( Y , τ ′ ) − → Ψ t o ( X , τ ) is defined b y Ψ t ( f ) ( F ) = f − 1 ( F ) , (61) for all F ∈ Ψ t o ( Y , τ ′ ). F urther, b y the pro o f of Theorem 2.1 1, ϕ is an DSk eLC - morphism. W e will sho w that ϕ satisfies condition (LO). W e ha v e that ϕ Λ : RC ( X ) − → R C ( Y ) is defined, according to (33) and (32 ) , by the form ula ϕ Λ ( F ) = cl( f ( F )), for ev ery F ∈ RC ( X ) . So, let F ∈ RC ( Y ), G ∈ C R ( X ) and F ρ Y ϕ Λ ( G ); then F ∩ f ( G ) 6 = ∅ and hence f − 1 ( F ) ∩ G 6 = ∅ ; therefore, ϕ ( F ) ρ X G . So, the axiom (LO) is fulfilled. Hence, Ψ t o ( f ) is an DOpLC - mo r phism. Therefore, t he con trav ari- an t functor Ψ t o is w ell defined. Let ϕ ∈ DOpLC (( A, ρ, I B) , ( B , η , I B ′ )). Put C = C ρ and C ′ = C η (see 1 .1 6 for the notations). Then, by 1.16, ( A, C ) and ( B , C ′ ) are CNCA’s. Set X = Ψ a o ( A, ρ, I B), Y = Ψ a o ( B , η , I B ′ ) a nd f = Ψ a o ( ϕ ). Then, by the pr o of of Theorem 2 .1 1, f : Y − → X is a contin uous sk eletal map. W e ar e no w going to sho w that f is an op en map. By (24), it is enough to pro v e that, for eve ry b ∈ I B ′ , f (int Y ( λ B ( b ))) is an op en subset of X (note tha t λ B ( b ) = λ g B ( b ) b ecause b ∈ I B ′ ). So, let b ∈ I B ′ . Let σ ∈ f (in t Y ( λ B ( b ))). Then there exists σ ′ ∈ int Y ( λ B ( b )) suc h that σ = f ( σ ′ ). By (42), b ∗ 6∈ σ ′ . Then 1.9 implies that there exists c 1 ∈ B such that b ∗ ≪ C ′ c ∗ 1 and c ∗ 1 6∈ σ ′ . Since σ ′ is a b ounded cluster in ( B , C ′ ), (38) implies that there exists c 2 ∈ I B ′ suc h that c ∗ 2 6∈ σ ′ . Put b 1 = c 1 ∧ c 2 . Then b 1 ∈ I B ′ ∩ σ ′ (b y (4 4)), b ∗ 1 6∈ σ ′ (b y (4 4)) a nd b ∗ ≪ C ′ b ∗ 1 (b y ( ≪ 3) (see 1.1)). Thu s b 1 ≪ C ′ b . Therefore, b y (42 ) and (4 3), σ ′ ∈ in t Y ( λ B ( b 1 )) ⊆ λ B ( b 1 ) ⊆ in t Y ( λ B ( b )). By 1.4, there exists u ∈ Ult ( B ) suc h that b 1 ∈ u ⊆ σ ′ and σ ′ = σ u . Put a = ϕ Λ ( b 1 ). Then, b y (4 0), a ∈ f ( σ ′ ) = σ . Supp ose tha t a ∗ ∈ σ . W e will sho w that this implies that ϕ ( a ∗ ) ∈ σ ′ . Indeed, supp ose that ϕ ( a ∗ ) 6∈ σ ′ . Then there exists c 3 ∈ u suc h that ϕ ( a ∗ )( − C ′ ) c 3 . Set b 2 = c 2 ∧ c 3 . Then b 2 ∈ u ∩ I B ′ and ϕ ( a ∗ )( − C ′ ) b 2 . Since C ′ = C η , w e obtain, by 1.16, that ϕ ( a ∗ )( − η ) b 2 . Using condition (LO), w e get that a ∗ ( − ρ ) ϕ Λ ( b 2 ). Since ϕ Λ ( b 2 ) ∈ I B (b y (L2)), w e obtain that a ∗ ( − C ) ϕ Λ ( b 2 ) ( see again 1.16). By (Λ1), ϕ ( ϕ Λ ( b 2 )) ≥ b 2 ; th us ϕ ( ϕ Λ ( b 2 )) ∈ u . Hence ϕ Λ ( b 2 ) ∈ ϕ − 1 ( u ). Since σ = f ( σ ′ ) = σ ϕ − 1 ( u ) and a ∗ ∈ σ , w e hav e that a ∗ C c , for ev ery c ∈ ϕ − 1 ( u ). Therefore a ∗ C ϕ Λ ( b 2 ), a con tra diction. Hence, ϕ ( a ∗ ) ∈ σ ′ , i.e. ( ϕ ( ϕ Λ ( b 1 ))) ∗ ∈ σ ′ . 21 Since, b y (Λ1), b ∗ 1 ≥ ( ϕ ( ϕ Λ ( b 1 ))) ∗ , w e obtain tha t b ∗ 1 ∈ σ ′ , a contradiction. Th us, a ∗ 6∈ σ . Then, using (4 2), (4 3) and (45), w e obta in that σ ∈ in t X ( λ A ( a )) ⊆ λ A ( a ) = λ A ( ϕ Λ ( b 1 )) = f ( λ B ( b 1 )) ⊆ f (int Y ( λ B ( b ))). Therefore, f (in t Y ( λ B ( b ))) is an op en set in X . Thu s, f is an op en map. Hence Ψ a o is well defined. F urther, note that, u sing (23), it is easy to see that λ g B is an DOpLC - isomorphism. The rest f o llo ws from Theorem 2.11. Definition 2.18 W e will denote b y OpC the catego ry of a ll compact Hausdorff spaces and all op en ma ps betw een them. Let DOpC b e the category whose ob jects are all complete normal con ta ct algebras and whose morphisms are all DSk eC -morphisms ϕ : ( A, C ) − → ( B , C ′ ) satisfying the follo wing condition: (CO) F or all a ∈ A and all b ∈ B , aC ϕ Λ ( b ) implies ϕ ( a ) C ′ b (see 1.21 fo r ϕ Λ ). It is easy to see that in this w ay we hav e defined categories. The category DOpC (resp., OpC ) is a (non-full) sub category of the category DSk eC (resp., Sk eC ). Theorem 2.19 The c ate gories OpC and DOpC ar e dual ly e quivalent. Pr o of. It follows from Theorem 2.17 and 1.15. Definition 2.20 Let OpPerLC b e the category of all lo cally compact Hausdorff spaces and all op en p erfect maps b etw een them. Let DOpPerLC b e the catego r y whose ob jects are all complete lo cal contact algebras (see 1.14) and whose morphisms a re all DSk ePerLC -morphisms satisfying condition (LO). It is easy to se e that in this w ay w e hav e defined categories. Ob viously , DOpP erLC (resp., OpP erLC ) is a sub cat ego ry of the category DSke P erLC (resp., SkeP erLC ). Theorem 2.21 The c ate gories OpP erLC and DOpP erLC ar e dual ly e q uivalent. Pr o of. It follows from Theorems 2.15 a nd 2.17. Note that since the morphisms of the category OpPerLC are closed maps, in the definition of the category DOpPe rLC (see 2.20) we can substitute condition (LO) for the follo wing one: (LO’) ∀ a ∈ A and ∀ b ∈ B , aρϕ Λ ( b ) implies ϕ ( a ) η b . 22 3 Connec ted Sp aces Notations 3.1 If K is a category whose ob jects form a sub class of the class of all top ological spaces (resp., con ta ct algebras) t hen we will denote b y KCon the full sub category of K whose ob jects are all “connected” K - ob jects, where “connected” is understo o d in the usual sense when the ob jects of K are top ological spaces and in the sense of 1.1 (see the condition (CON) t here) when the ob jects of K are contact algebras. F or example, w e denote b y: • SkeP erLCCon the full sub category of the category Sk ePerLC ha ving a s o b jects all connected lo cally compact Hausdorff spaces; • DSk eP erLCCon t he f ull subcatego r y of the category DSkeP erLC ha ving as ob jects all connected CLCA’s. Theorem 3.2 The c ate gories SkeP erLCCon and DSk ePerLCCon ar e dual ly e quivalent; in p articular, the c ate gories Sk eCCon and DSk eCCon ar e dual ly e quiv- alent. Pr o of. It follows immediately fr o m 1.11, Theorem 2.15 and Theorem 2.13. Theorem 3.3 The c ate gories OpP erLCCon and DOpP erLCCon ar e dual ly e qu- ivalent; in p articular, the c ate gories OpCCon and DOpCCon ar e dual ly e quiva- lent. Pr o of. It follows immediately fr o m 1.11, Theorem 2.19 and Theorem 2.21. Analogously one can formulate and pro ve the connected v ersions of the theo- rems Theorem 2.1 1 and Theorem 2.17. 4 Equiv alence The o rems Definition 4.1 ([11]) Let ESkeC b e the category whose ob jects are all complete normal con ta ct algebras and whose morphisms ψ : ( A, C ) − → ( B , C ′ ) are all func- tions ψ : A − → B satisfying the follow ing conditions: (EF1) for eve ry a ∈ A , ψ ( a ) = 0 iff a = 0; (EF2) ψ preserv es all joins; (EF3) if a ∈ A , b ∈ B and b ≤ ψ ( a ) then there exists c ∈ A suc h that c ≤ a and ψ ( c ) = b ; (EF4) for eve ry a, b ∈ A , aC b implies that ψ ( a ) C ′ ψ ( b ). In [11], V. V. F edorch uk prov ed the f o llo wing theorem: Theorem 4.2 ([11]) The c a te gories Sk eC and ESk eC ar e e q uiva lent. W e will no w prese n t a generalization of this theorem. 23 Definition 4.3 Let ESke LC b e the category whose ob jects are all complete lo cal con ta ct algebras and whose morphisms ψ : ( A, ρ, I B) − → ( B , η , I B ′ ) are a ll functions ψ : A − → B satisfying conditions (EF1 ) - (EF3) (see D efinition 4.1) and the following t wo constrain ts: (EL4) for ev ery a, b ∈ A , aρb implies that ψ ( a ) η ψ ( b ); (EL5) if a ∈ I B then ψ ( a ) ∈ I B ′ . The pro of of the follow ing theorem is similar to t ha t of The orem 4.2. Theorem 4.4 The c ate gories Sk eLC and ESk eLC ar e e quivalent. Pr o of. Since the categories Sk eLC a nd DSkeL C are dually equiv alen t ( by Theorem 2.11), it is enoug h to sho w that the categories E Sk eLC and DSk eLC are dually equiv alen t. Let us define a contra v arian t functor D p : ESkeLC − → DSkeLC . Let D p b e the iden tity on the ob jects of the category ESk eLC and let, f or every ψ ∈ ESk eLC (( A, ρ, I B) , ( B , η , I B ′ )), D p ( ψ ) = ψ P , where ψ P is the right adjoint o f ψ (see 1.21 and (EF2)). Setting ϕ = ψ P , w e hav e to sho w that ϕ ∈ DSkeL C (( B , η , I B ′ ) , ( A, ρ, I B)) . As it is prov ed in [11], ϕ is a complete Bo olean homomorphism. F or complete- ness of our exp osition, w e will presen t here the F edorc h uk’s pro of. Note first that ψ = ϕ Λ . By 1.2 1, ϕ preserv es all meets in B . Since, b y (EF1), ψ (0) = 0 , w e ha ve that ϕ (0) = ϕ ( ψ (0)); if ϕ (0) > 0 then, b y (EF1) and 1.21, 0 = ψ (0) = ψ ( ϕ ( ψ (0)) ) > 0, a con tradiction. Hence ϕ (0) = 0. F urther, since ψ (1) ≤ 1 ⇐ ⇒ 1 ≤ ϕ (1), w e get that ϕ (1) = 1 . Finally , ϕ ( b ∗ ) = ( ϕ ( b )) ∗ , for ev ery b ∈ B . Indeed, let b ∈ B . Set a = ϕ ( b ) ∧ ϕ ( b ∗ ). Then, by 1.21, ψ ( a ) ≤ ψ ( ϕ ( b )) ∧ ψ ( ϕ ( b ∗ )) ≤ b ∧ b ∗ = 0. Henc e ψ ( a ) = 0. Therefore, b y (EF1), a = 0, i.e. ϕ ( b ) ∧ ϕ ( b ∗ ) = 0 . Set no w c = ϕ ( b ) ∨ ϕ ( b ∗ ) and supp ose that c < 1. Then c ∗ 6 = 0 . Since 0 = c ∗ ∧ c = ( c ∗ ∧ ϕ ( b )) ∨ ( c ∗ ∧ ϕ ( b ∗ )), w e hav e that c ∗ ∧ ϕ ( b ) = 0 = c ∗ ∧ ϕ ( b ∗ ). By (EF1), ψ ( c ∗ ) 6 = 0. Ob viously , ψ ( c ∗ ) = ( ψ ( c ∗ ) ∧ b ) ∨ ( ψ ( c ∗ ) ∧ b ∗ ). Therefore, at least one of the elemen ts ψ ( c ∗ ) ∧ b and ψ ( c ∗ ) ∧ b ∗ is differen t from 0. Let ψ ( c ∗ ) ∧ b 6 = 0. By ( EF3), the inequal- it y ψ ( c ∗ ) ∧ b ≤ ψ ( c ∗ ) implies that there exists d ∈ A suc h that d ≤ c ∗ and ψ ( d ) = ψ ( c ∗ ) ∧ b . Since ψ ( d ) 6 = 0, w e get, b y (EF1), that d 6 = 0. F urther, ψ ( d ) ≤ b implies that d ≤ ϕ ( b ). Then d ≤ c ∗ ∧ ϕ ( b ) = 0, i.e. d = 0, a con tradiction. Analo- gously , w e obtain a contradiction if ψ ( c ∗ ) ∧ b ∗ 6 = 0. So, c = 1, i.e. ϕ ( b ) ∨ ϕ ( b ∗ ) = 1 . Hence, w e hav e prov ed that ϕ ( b ∗ ) = ( ϕ ( b )) ∗ . All this sho ws that ϕ is a complete Bo olean homomorphism. Since conditions (L1) and (EL1) in 2.10 are equiv alen t and ψ = ϕ Λ , (EL4) implies that ϕ satisfies condition (L1). Ob viously , ( EL5 ) implies tha t ϕ satisfies condition (L2) in 2.10. So, ϕ is a DSk eLC -morphism. Now , from D p ( id ) = id and the formu la ( ψ 2 ◦ ψ 1 ) P = ( ψ 1 ) P ◦ ( ψ 2 ) P , we obta in that D p is a contra v ariant f unctor . Let us define a contra v ariant functor D l : DSkeL C − → ESkeL C . Let D l b e the identit y on t he o b jects of the catego ry DSk eLC and let, for ev ery ϕ ∈ 24 DSk eLC (( A, ρ, I B) , ( B , η , I B ′ )), D l ( ϕ ) = ϕ Λ , where ϕ Λ is the left adjoin t of ϕ (see 1.21). Setting ψ = ϕ Λ , w e hav e to sho w that ψ ∈ E Sk eLC (( B , η , I B ′ ) , ( A, ρ, I B)) . Since 0 ≤ ϕ ( 0) implies that ψ (0) ≤ 0, w e get tha t ψ (0) = 0. If ψ ( b ) = 0 then ψ ( b ) ≤ 0 and hence b ≤ ϕ (0) = 0, i.e. b = 0. Therefore, ψ satisfies condition (EF1 ) . F urther, conditions (EF2), (EL4) a nd (EL5) are clearly satisfied b y ψ . Finally , let a ≤ ψ ( b ). Set c = b ∧ ϕ ( a ). Then c ≤ b and, by 1.22(b), ψ ( c ) = a ∧ ψ ( b ) = a . Therefore, ψ satisfies condition (EF3 ). So, ψ is an ESk eLC -morphism. No w, it is clear t ha t D l is a con trav arian t functor. Since the comp o sitions of D p and D l are the iden tity functors, w e get that D p is a duality . Put no w Φ a = Ψ a ◦ D p and Φ t = D l ◦ Ψ t . Then Φ a : ESk eLC − → Sk eLC and Φ t : Sk eLC − → ESk eLC are the required equ iv alences. Definition 4.5 Let ESkeP erLC b e the category whose ob jects are a ll complete lo cal contact algebras (se e 1.14) and whose mo r phisms are a ll ESk eLC -morphisms ψ : ( A, ρ, I B) − → ( B , η , I B ′ ) satisfying the fo llo wing condition: (EL6) if b ∈ I B ′ then ψ P ( b ) ∈ I B (where ψ P is the rig h t adjoin t of ψ (see 1.21) ) . Theorem 4.6 The c ate gories Sk eP erLC and E SkeP erLC ar e e quivalent. Pr o of. Using Theorem 2.15, it is enough to sho w that the categories DSk eP erLC and ESkeP erLC are dually equiv alen t . W e will sho w that the restriction of the con trav arian t functor D p (defined in the pro o f of Theorem 4.4) to the category ESk ePerLC is the required duality functor. Let ψ ∈ E Sk ePerLC (( A, ρ, I B) , ( B , η , I B ′ )). Then, b y (EL6), ψ P satisfies con- dition (L3) in 2.14. Hence, b y the pro of of Theorem 4.4, D p ( ψ ) is a DSk eP erLC - morphism. F urther, let us regar d the restriction of the con tra v arian t functor D l (defined in the pro of of Theorem 4.4) to the category DSk ePerLC . If ϕ is a DSk ePerLC -morphism then, b y (L3), ϕ Λ satisfies condition (EL6). Hence D l ( ϕ ) is an ESk ePerLC - morphism. Therefore, D p is a dualit y . Definition 4.7 Let EOpLC b e the category whose ob jects are all complete lo cal con ta ct algebras and whose morphisms are all ESk eLC -morphisms ψ : ( A, ρ, I B) − → ( B , η , I B ′ ) satisfying the fo llo wing condition: (EL7) if b ∈ B , a ∈ I B a nd ψ ( a ) η b then aρψ P ( b ) (where ψ P is t he rig h t adjoin t of ψ (see 1.21 ) ). Theorem 4.8 The c ate gories OpLC and EOpLC ar e e quivalent. Pr o of. It is clear tha t if ψ satisfies condition (EL7) then ψ P satisfies condition (LO) in 2.16 and if ϕ satisfies condition (LO ) then ϕ Λ satisfies (EL7). No w, using Theorem 2.17 , w e a rgue as in the pro of of Theorem 4.6. 25 Definition 4.9 Let E OpC b e the catego r y whose ob jects a re all complete no rmal con ta ct algebras and whose morphisms ar e all E Sk eC -mo r phisms ψ : ( A, C ) − → ( B , C ′ ) satisfying the fo llo wing condition: (EC7) if a ∈ A , b ∈ B and ψ ( a ) C ′ b then a C ψ P ( b ) (where ψ P is the righ t adjoin t of ψ ( see 1.21)). Theorem 4.10 The c ate gories OpC and EOpC ar e e quivalent. Pr o of. It follows directly f rom Theorem 4 .8 . Definition 4.11 Let EOpP erLC b e the category whose ob jects ar e all complete lo cal con tact algebras and whose morphisms are all ESk eP erLC -morphisms satis- fying condition (EL7). Theorem 4.12 The c ate gories OpP erLC and EOpP erLC ar e e q uivalent. Pr o of. It follows from the pro o fs of 4.6 and 4.8. Remark 4.13 A great pa rt of our Theorem 4.4 is form ulated (in another for m) and pro v ed in Ro ep er’s paper [20]. Let us state precis ely what is done there (using our notations). Ro eper defines the notion of mer e olo gi c a l mapping : such is any function ψ : B − → A , where A and B are complete Bo olean algebras, whic h satisfies the follo wing conditions: (i) ψ ( b ) = 0 iff b = 0 ; (ii) a ≤ b implies ψ ( a ) ≤ ψ ( b ); (iii) if 0 6 = a ≤ ψ ( b ), where b ∈ B and a ∈ A , then there exists b ′ ∈ B suc h that 0 6 = b ′ ≤ b and ψ ( b ′ ) ≤ a . It is sho wn that an y mereological mapping preserv es all joins in B . F urther, a mapping ψ of a CLCA ( B , η , I B ′ ) to another CLCA ( A, ρ, I B) is called: (a) c o n tinuous if aη b implies ψ ( a ) ρψ ( b ), and (b) b ounde d if ψ ( b ) ∈ I B when b ∈ I B ′ . It is shown that ev ery contin uous a nd b ounded mereological mapping ψ : ( B , η , I B ′ ) − → ( A, ρ, I B) generates a f unction f ψ : Ψ a ( B , η , I B ′ ) − → Ψ a ( A, ρ, I B), defined by the formula f ψ ( σ u ) = σ ψ ( u ) , for ev ery u ∈ Ult( B ); the f unction f ψ is con tinuous (in top ological sense) and is suc h tha t cl( f ψ ( F )) is regular closed when F is regular closed. It is pro ve d that if f : Ψ a ( B , η , I B ′ ) − → Ψ a ( A, ρ, I B) is a con tinuous function suc h that cl( f ( F )) is regular closed when F is regular closed then there exists a con tin uous and b ounded mereological function ψ : ( B , η , I B ′ ) − → ( A, ρ, I B) suc h that f = f ψ . Finally , a mereological function ψ : ( B , η , I B ′ ) − → ( A, ρ, I B) is called top olo gic al if ψ (1 B ) = 1 A , ψ ( a ) ρψ ( b ) iff aη b , and ψ ( b ) ∈ I B iff b ∈ I B ′ ; it is sho wn that if ψ is top ological then f ψ is a ho meomorphism and if f : Ψ a ( B , η , I B ′ ) − → Ψ a ( A, ρ, I B) is a ho meomorphism then there exists a top ological function ψ : ( B , η , I B ′ ) − → ( A, ρ, I B) suc h that f = f ψ . It is easy to see that a function ψ : B − → A is mereological iff it satisfies conditions (EF1)-(EF3) (see Definition 4.1); ψ is contin uous (resp ective ly , b ounded) iff it satisfies condition (EL4) (resp ectiv ely , ( EL5)). F urther, Lemma 2 .6 sho ws that a con tin uous map f : X − → Y satisfies Ro eper’s condition “cl( f ( F )) ∈ RC ( Y ) 26 when F ∈ RC ( X )” iff f is a sk eletal map. 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