Some Generalizations of Fedorchuk Duality Theorem -- II

Some Generalizations of F edorc h uk Dualit y Theorem – I I Georgi Dimo v ∗ Dept. of Math. and Informatics, Sofia Univ ers it y , Blvd. J. Bourchier 5, 1164 Sofia, Bulg aria Abstract This pap er is a con tin ua tion of t he pap er [4]. In [4] it w as sho wn that there exists a dualit y Ψ a betw een the category D Sk eLC ( int ro duced there) and the category Sk eLC of lo c a lly compact Hausdorff spaces and contin uous skeletal ma ps. W e de- scrib e here the subca tegories of the category DSk eLC which are dually equiv a lent to the following e ight ca tegories : all o f them hav e as ob jects the lo c ally co mpact Hausdorff spaces and their morphisms ar e, resp ectively , the injective (resp ectively , surjective) co n tin uous skeletal maps, the injectiv e (surjective) o p e n maps, the injec- tive (sur jective) s keletal p er fect maps, the injectiv e (surjective) open p e rfect maps. The pa rticular cases of these theo r ems for the full sub categor ies of the last four ca t- egories having as ob jects all compact Hausdorff spa ces are formulated and pr ov ed. The DSk eLC -morphisms whic h are LCA-embedding s and the dens e homeomorphic embeddings ar e characterized through their dual mo rphisms. F or any loca lly com- pact space X , a description of the frame o f all op en subsets o f X in ter ms of the dual ob ject of X is obtained. It is shown ho w one can build the dual ob ject o f an o p en subset (respec tively , of a r egular clo sed subset) of a lo c a lly co mpa ct Hausdor ff space X directly from the dual ob ject of X . Applying these results, a new descr iption of the ordered set of all, up to equiv alence, lo cally co mpa ct Hausdorff extensions of a lo cally compact Hausdorff space is obtained. Mor e ov er , genera lizing de V rie s Com- pactification Theorem ([2]), we strengthen the Lo cal Compactifica tion Theor e m of Leader ([10]). So me o ther applications are found. MSC: primary 54D45, 1 8A40; secondar y 54C10 , 54D35, 54E05. Keywor ds: Norma l contact algebra; Loca l contact algebra; Locally compact (compact) space; Skeletal map; (Quasi-)Op en p erfect ma p; Open ma p; Injective (surjective) mapping; Embedding ; Dualit y; F rame; Lo cally compact (compact) e x tension. ∗ This pap er was supp o rted by the pro ject no. 1 01/20 07 “Categorical T opolo gy” of the Sofia Univ ersity “St. Kl. Ohridski”. 1 E-mail addr ess: gdimo v@fmi.uni-sofia.bg 1 In tro duct ion This pap er is the second part o f the pap er [4]. In [4], the category D SkeLC of all complete lo cal con tact algebras and all complete Bo olean homomorphisms b et w een them satisfying tw o simple conditions (see [4, Definition 2.10]) w as defined and it w as show n that there exists a dualit y Ψ a : DSkeLC − → SkeL C , where SkeLC is the category of all lo cally c ompact Hausdorff spaces and all sk eletal (in the sense of Mio duszewski and Rudolf [11]) con tin uous maps b etw een them. In the first section of the presen t pap er, we find the sub categories of the category DSkeL C whic h are dually equiv alen t to the f ollo wing eight sub categor ies of the category Sk eLC : all of them ha ve as ob jects the lo cally compact Hausdorff spaces a nd their morphisms are, resp ectiv ely , the injectiv e ( r esp ective ly , surjectiv e) con t in uous sk eletal maps (see Theorem 1.6 (resp., T heorem 1.3)), the injectiv e (surjectiv e) op en maps (s ee Theorem 1 .12 (resp., Theorem 1.13)), the injectiv e (surjectiv e) sk eletal p erfect maps (see Theorem 1.10 (resp., Theorem 1.8)), the injectiv e ( surjectiv e) op en p erfect maps (see Theorem 1 .15 (resp., Theorem 1.16)). In the theorems men tio ned ab o ve , the particular cases f or the full sub categories of the last four catego ries ha ving as ob jects all compact Hausdorff spaces are form ulated as w ell. In the second section, w e pro v e that ϕ is a DSk eLC -morphism and an LCA- em b edding iff f = Ψ a ( ϕ ) is a quasi-op en se mi-op en p erfect surjection (see Theorem 2.4). The dens e homeomorphic em b eddings are c haracterized through their dual morphisms as well (see Theorem 2.2). F or any lo cally compact Hausdorff space X , a description of the f r ame of all op en subsets o f X in terms of the dual to X complete lo cal con tact algebra is o btained (see Theorem 2 .8 ). It is sho wn ho w one can build the dual ob ject of an op en subset (respectiv ely , of a regular closed subset) of a lo cally compact Hausdorff space X directly from the CLCA dual to X (see Theorem 2.9 (resp ectiv ely , Theorem 2.10)). The third section is dev oted to some immediate a pplications of the theorems obtained ab ov e. A new description of the ordered set o f all (up to equiv alence) Ha us- dorff locally compact extensions of a lo cally compact Hausdorff space is obtained in the la ng uage of normal contact relations (see Theorem 3.11). The normal contact relations whic h corresp o nd to the Alexandroff (one-p oin t) compactification a nd t o the Stone- ˇ Cec h compactification of a lo cally compact Hausdorff space are found (see Theorem 3.12). The W allma n- t ype compactifications of discrete spaces are described as well (see Theorem 3 .25). G eneralizing de V r ies Compactification Theorem ([2]) (see Coro lla ry 3.2 3), a theorem is obt a ined (see Theorem 3.20) whic h strengthen the Lo cal Compactification Theorem of Leader ([10]). Some other applications are found. In this pap er w e use the notations in tro duced in the first part of it (see [4]), as w ell a s the notions and the results from [4]. If A is a Bo olean algebra then w e denote by Atoms ( A ) t he set of all a toms of A . If ( A, ≤ ) is a p oset and a ∈ A then ↓ a is the set { b ∈ A | b ≤ a } . If f : X − → Y is a function and M ⊆ X then f ↾ M is the restriction o f f ha ving M as a domain and f ( M ) as a co do ma in. Finally , we 2 will denote by I D the set o f a ll dy a dic n um b ers of the interv al (0 , 1). 1 Surjectiv e and In jectiv e Mapping s Notations 1.1 W e denote by: • I nSk eLC the category of all locally compact Hausdorff spaces and all con tin uous sk eletal injectiv e maps b etw een them; • SuSk eLC the category of all locally compact Hausd orff s paces and a ll con tin uous sk eletal surjectiv e maps b etw een them. In the sequel w e will use similar notations without more explanations, i.e. if K is a category introduced in [4], then b y InK (resp., SuK ) w e will denote the category ha ving the same ob jects as the category K and whose morphisms are only the injectiv e (resp., surjectiv e) mo r phisms of K . Definition 1.2 Let DSuSk eLC b e the category whose ob jects are all complete lo- cal con t a ct algebras (see [4, 1.14]) and whose mor phisms are all DSkeLC -morphisms (see [4, 2.10]) ϕ : ( A, ρ, I B) − → ( B , η , I B ′ ) whic h satisfy the following condition: (IS) F or ev ery b ounded ultrafilter u in ( A, ρ, I B) there exists a b ounded ultrafilter v in ( B , η , I B ′ ) suc h that ϕ Λ ( v ) ρu (see [4, 1.18] and [4, 1.21 ] for the notations). Using [4, Corollary 1.5, (EL1), (36), 1.21], we obtain easily that DSuSk eLC is indeed a category . Theorem 1.3 The c ate gories SuSk eLC and DSuSk eLC ar e dual ly e quivalent. Pr o o f. Let f : X − → Y b e a surjectiv e contin uous sk eletal map betw een t wo lo cally compact Ha usdorff spaces and ϕ = Ψ t ( f ). Then ϕ : RC ( Y ) − → RC ( X ) and ϕ Λ ( F ) = cl( f ( F )), fo r ev ery F ∈ RC ( X ) (see the pro of of [4, Theorem 2.11]). Let u b e a b ounded ultrafilter in RC ( Y ). Then there exists G 0 ∈ C R ( Y ) ∩ u . Hence there exists y ∈ T { G | G ∈ u } . Since f is a surjection, there exists x ∈ X suc h that f ( x ) = y . Let v b e an ultrafilter in RC ( X ) whic h con tains ν x (see [4, (3) ] for ν x ). Then, obv iously , v is a b ounded ultrafilter in ( RC ( X ) , ρ X , C R ( X )). By [4, (51)], v ⊆ σ x (see [4, (3)] fo r σ x ). Hence y ∈ ϕ Λ ( F ), for ev ery F ∈ v . This means that ϕ Λ ( v ) ρ Y u . Therefore, ϕ satisfies condition ( IS). By [4, Theorem 2 .1 1], ϕ is a DSk eLC -morphism. Hence, ϕ is a DSuSk eLC -morphism. Let ϕ : ( A, ρ, I B) − → ( B , η , I B ′ ) b e a DSuSk eLC -morphism and f = Ψ a ( ϕ ). Let X = Ψ a ( A, ρ, I B), Y = Ψ a ( B , η , I B ′ ) and σ ∈ X . The n σ is a b o unded cluster in ( A, C ρ ). Hence there exists a b ounded ultrafilter u in ( A, ρ, I B) s uc h that σ = σ u . By (IS), there exists a b o unded ultr a filter v in ( B , η , I B ′ ) such that ϕ Λ ( v ) ρu . Th us ϕ Λ ( v ) C ρ u . Therefore, by [4, 1.5, (36), (37)], and [4, ( 3 5)], f ( σ v ) = σ ϕ Λ ( v ) = σ u = σ . So, f is a surjection. By [4, Theorem 2.11], f is a con tinuous sk eletal ma p. Hence , f is a SuSkeL C -morphism. The rest follows from [4, Theorem 2.11]. 3 Corollary 1.4 Every DSuSk eLC -mo rp h ism is an inje ction. Pr o o f. Let ϕ : ( A, ρ, I B) − → ( B , η , I B ′ ) b e a DSuSk eLC -morphism. Set f = Ψ a ( ϕ ), X = Ψ a ( B , η , I B ′ ) and Y = Ψ a ( A, ρ, I B). Then, b y Theorem 1.3, f : X − → Y is a surjectiv e con tin uous sk eletal map. Let ˆ ϕ = Ψ t ( f ). Then ˆ ϕ : RC ( Y ) − → RC ( X ). W e will sho w that ˆ ϕ is an injection. W e ha v e that, for ev ery F ∈ R C ( Y ), ˆ ϕ ( F ) = cl(in t ( f − 1 ( F ))) = cl( f − 1 (in t ( F ))). Let F , G ∈ RC ( Y ) and ˆ ϕ ( F ) = ˆ ϕ ( G ). Supp ose that there exists a p oin t x ∈ F \ G . Then there exists an op en neigh b orho o d O x of x suc h that O x ∩ G = ∅ . Obv iously , there exists a p oint y ∈ O x ∩ in t( F ). Then f − 1 ( y ) ∩ f − 1 ( G ) = ∅ . But f − 1 ( y ) ⊆ f − 1 (in t( F )) ⊆ ˆ ϕ ( F ) = ˆ ϕ ( G ) ⊆ f − 1 ( G ) and th us f − 1 ( y ) = ∅ . Since f is a surjection, w e obtain a con tradiction. Hence F ⊆ G . Analogously w e pro ve that G ⊆ F . Therefore F = G . So, ˆ ϕ is an injection. No w, using Theorem 1.3, w e get that ϕ is an injection. Definition 1.5 Let DI nSk eLC b e the category whose o b jects are all complete lo- cal con t a ct algebras (see [4, 1.14]) and whose mor phisms are all DSkeLC -morphisms (see [4, 2.10]) ϕ : ( A, ρ, I B) − → ( B , η , I B ′ ) whic h satisfy the following condition: (LS) ∀ a, b ∈ I B ′ , ϕ Λ ( a ) ρϕ Λ ( b ) implies aη b (see [4, 1.21] for ϕ Λ ). It is easy to see that DInSk eLC is indeed a category . Theorem 1.6 The c ate gories InSke LC and DInSk eLC ar e dual ly e quivalent. Pr o o f. Let f : X − → Y b e an injectiv e con tinuous sk eletal map b etw een t w o lo cally compact Hausdorff spaces X and Y . Set ϕ = Ψ t ( f ) (see [4, (31)] f o r Ψ t ( f )). The function ϕ Λ : RC ( X ) − → R C ( Y ) is defined b y ϕ Λ ( F ) = cl( f ( F )), for ev ery F ∈ RC ( X ) (see [4, (32) and (33)]). Hence, fo r F ∈ C R ( X ), ϕ Λ ( F ) = f ( F ). Since f is a n injection, it b ecomes o bvious that ϕ satisfies condition (LS) from 1.5. By [4, Theorem 2.11 ], Ψ t ( f ) is a DSk eLC - morphism. Thus w e g et that Ψ t ( f ) is a DInSk eLC -morphism. Let ϕ : ( A, ρ, I B) − → ( B , η , I B ′ ) b e a morphism of the category DInSk eLC and let f = Ψ a ( ϕ ) (see [4, (35)] for Ψ a ( ϕ )). W e will sho w that f is an injection. Let σ and σ ′ b e tw o b ounded clusters in ( B , C η ) (see [4, 1.16 ] for C η ) and σ 6 = σ ′ . Since Y = Ψ a ( B , η , I B ′ ) is a lo cally compact Hausdorff space, w e obtain, using [4, (24)], that there exist b, b ′ ∈ I B ′ suc h that σ ∈ in t( λ g B ( b )), σ ′ ∈ in t ( λ g B ( b ′ )) and λ g B ( b ) ∩ λ g B ( b ′ ) = ∅ . Th us, by [4, 2.1 ] and [4, (14)], b ( − η ) b ′ . No w, (LS) implies that ϕ Λ ( b )( − ρ ) ϕ Λ ( b ′ ). Using condition (L 2 ) from [4], we get tha t ϕ Λ ( b ) , ϕ Λ ( b ′ ) ∈ I B. Hence ϕ Λ ( b )( − C ρ ) ϕ Λ ( b ′ ). Since, by [4, (40)], ϕ Λ ( b ) ∈ f ( σ ) and ϕ Λ ( b ′ ) ∈ f ( σ ′ ), w e obtain that f ( σ ) 6 = f ( σ ′ ). So, f is an injection. Fina lly , b y [4, Theorem 2.11], f is a con tinuous sk eletal map. Th us f is an InSk eLC -morphism. No w, [4, Theorem 2.11] implies that the categories InSk eLC and DInSk eLC are dually equiv alen t. 4 Notations 1.7 W e denote by: • DSuSk ePerLC the categor y of a ll CLCA’s and all injectiv e complete Bo olean homomorphisms b et we en them satisfyin g axioms (L1) -(L3) (see [4 , Definitions 2.10, 2.14]); • DSuSk eC the category o f all CNCA’s and all injectiv e complete Bo olean homo- morphisms b etw een them satisfying axiom (F1) (see [4, Definition 2.12]). Theorem 1.8 The c ate gori e s SuSk eP erLC and DSuSke P erLC ar e dual ly e quiv- alent; in p articular, the c ate gories SuSk eC and D SuSkeC ar e dual ly e quivalent. Pr o o f. Let f : X − → Y b e a surjectiv e sk eletal p erfect map b et we en t w o lo cally compact Hausdorff spaces X and Y . Then the pro of of Corollary 1.4 sho ws t ha t Ψ t ( f ) : RC ( Y ) − → RC ( X ) is a n injection. By [4 , Theorem 2.1 5], Ψ t ( f ) is an DSk ePerLC -morphism. So, we get that Ψ t ( f ) is a DSuSk ePerLC -morphism. Let ϕ : ( A, ρ, I B) − → ( B , η , I B ′ ) b e a DSuSk ePerLC -morphism. Then ϕ is an injection. Hence ϕ Λ ( ϕ ( a )) = a , for ev ery a ∈ A . W e will show tha t ϕ satisfies condition ( IS) from 1 .2 . Let u b e a b ounded ultrafilter in ( A, ρ, I B). Then there exists a ∈ u ∩ I B. Since ϕ is an injection, ϕ ( u ) is a filter base in B . Hence, there exists v ∈ Ult( B ) suc h that ϕ ( u ) ⊆ v . By the condition (L 3 ) from [4 ], ϕ ( a ) ∈ I B ′ . Therefore, v is a b ounded ultra filter in ( B , η , I B ′ ). Moreov er, ϕ Λ ( v ) = u . Indeed, since ϕ ( u ) ⊆ v and ϕ Λ ◦ ϕ = id A , w e get that u ⊆ ϕ Λ ( v ). F urther, using [4, 1.21], w e obtain easily that ϕ Λ ( v ) is a filter base. Hence, ϕ Λ ( v ) = u . Therefore ϕ Λ ( v ) ρu . Th us ϕ satisfies condition (IS). Then Theorem 1.3 implies t ha t f = Ψ a ( ϕ ) is a surjection. By [4, Theorem 2.15], f is a sk eletal p erfect map. So, w e get that f is a SuSk ePerLC -morphism. No w, using [4, Theorem 2.1 5], w e obtain that the categories SuSkeP erLC and DSuSk eP erLC are dually equiv alen t. In pa rticular, the categories SuSk eC and DSuSk eC are dually equiv alen t. Definition 1.9 Let DInSk eP erLC b e t he category whose ob j ects a re all complete lo cal con tact algebras (see [4, 1 .14]) and whose morphisms are all D Sk ePerLC - morphisms (see [4, 2.14 ]) whic h satisfy condition (LS) (see 1.5). Let DInSk eC b e the category whose ob jects are all complete normal con tact algebras (see [4, 1.1]) and whose morphisms are a ll DSkeC -morphisms (see [4, 2.12]) ϕ : ( A, C ) − → ( B , C ′ ) whic h satisfy the following condition: (FS) ∀ a, b ∈ B , ϕ Λ ( a ) C ϕ Λ ( b ) implies aC ′ b . Theorem 1.10 The c ate gories InSk ePe rLC and DInSk eP erLC ar e dual ly e quiv- alent; in p articular, the c ate gories InSk eC a n d DI nSk eC ar e d ual ly e quiva lent. Pr o o f. It follows from [4, Theorem 2.15] and Theorem 1.6. Note that s ince the morphis ms of the category I nSk ePerLC are closed maps, in the definition of the category DI nSkeP erLC (see 1.9) w e can substitute condition (LS) for the fo llo wing one: 5 (ELS) ∀ a, b ∈ B , ϕ Λ ( a ) ρϕ Λ ( b ) implies aη b (see [4, 1.21] for ϕ Λ ). Notations 1.11 W e denote by: • DSuOpLC the categor y of all CLCA’s and all complete Bo olean homomorphisms b et w een t hem satisfying axioms (L1), (L2), (IS) and (LO) (see [4, 2.10], 1.2 a nd [4, 2.16]); • DInOpLC the category of all CLCA’s a nd all surjectiv e complete Bo olean homo- morphisms b etw een them satisfying axioms (L1), (L2) and (LO) (see [4, 2.10] and [4, 2.16]). Theorem 1.12 The c ate gories InOpLC a n d DI nOpLC ar e dual ly e quivalent. Pr o o f. Let us sho w that eve ry DInOpLC -mor phism is a DInSk eLC -morphism. Indeed, let ϕ : ( A, ρ, I B) − → ( B , η , I B ′ ) b e a morphism of the category DI nOpLC . Then ϕ is a surjection. Let a, b ∈ I B ′ and ϕ Λ ( a ) ρϕ Λ ( b ). Then, b y condition (LO) from [4], ϕ ( ϕ Λ ( a )) η b . Hence, by [4, 1 .21], aηb . Therefore, ϕ satisfies condition (LS). Hence ϕ is a DI nSkeLC -morphism. Let f : X − → Y be an InOpLC -morphism and ϕ = Ψ t ( f ). Then ϕ ( G ) = f − 1 ( G ), for ev ery G ∈ RC ( Y ) (see the pro o f o f [4 , Theorem 2.17 ]). F or ev ery F ∈ RC ( X ) w e ha ve , b y [4, Corollary 2.5 ] and [4, Lemma 2.6], that cl( f ( F )) ∈ RC ( Y ). Set G = cl( f ( F )). Then, by [6, 1.4.C], f − 1 ( G ) = cl( f − 1 ( f ( F ))) (b ecause f is an op en map), and the injectivit y of f implies that f − 1 ( G ) = F . Hence ϕ ( G ) = F . Therefore, ϕ is a surjection. No w, our theorem follows from Theorem 1.6 and [4, Theorem 2.17]. Theorem 1.13 The c ate gories SuOpLC and D SuOpLC ar e d ual ly e quivale n t. Pr o o f. It follows from Theorem 1.3 and [4, Theorem 2.17]. Notations 1.14 W e will denote by: • D SuOpPerLC the category of all CLCA’s and all injectiv e complete Bo olean homomorphisms b et w een them satisfying axioms (L1)-(L3) and (LO) (see [4, Defi- nitions 2.10, 2.14, 2.1 6]); • DI nOpPerLC the category of all CLCA’s and all surjectiv e complete Bo o lean homomorphisms b et w een them satisfying axioms (L1)-(L3) and (LO) (see [4, Defi- nitions 2.10, 2.14, 2.1 6]); • DSuOpC the categor y of all CNCA’s and all injectiv e complete Bo olean homo- morphisms betw een them s atisfying axioms (CO) and (F1) (se e [4, Definitions 2.12, 2.18]); • DInOpC the cat ego ry of all CNCA’s and all surjectiv e complete Bo o lean homo- morphisms betw een them s atisfying axioms (CO) and (F1) (se e [4, Definitions 2.12, 2.18]). 6 Theorem 1.15 The c ate gories InOpPe rLC a n d DInOpPerLC ar e dual ly e quiv- alent; in p articular, the c ate gories InOpC and DInOpC ar e dual ly e quivalent. Pr o o f. It follows from Theorem 1.12 and [4, Theorem 2.21]. Theorem 1.16 The c ate gories SuOpP erLC and DSuOpP erLC ar e dual ly e quiv- alent; in p articular, the c ate gories SuOpC and DSuOpC ar e dual ly e quivalent. Pr o o f. It follows from Theorem 1.8 and [4, Theorem 2.21]. 2 Em b eddings . Op en sets . Regular c l o sed sets W e will need a lemma from [1]: Lemma 2.1 L et X b e a dense subsp ac e of a top olo gic al sp ac e Y . Then the functions r : RC ( Y ) − → RC ( X ) , F − → F ∩ X , and e : R C ( X ) − → RC ( Y ) , G − → cl Y ( G ) , ar e Bo ole an isomorph isms b etwe e n B o ole an algebr as RC ( X ) and RC ( Y ) , and e ◦ r = id RC ( Y ) , r ◦ e = id RC ( X ) . Theorem 2.2 If a function f b e twe en two lo c al ly c omp act Hausdorff sp ac es is a dense home omorphic emb e dding then ϕ = Ψ t ( f ) is a DInOpLC -morphism and a Bo ole an isomorphism. Con v ersely, if a function ϕ is a DI nOpLC -morphism and a Bo ole an is o morphism then f = Ψ a ( ϕ ) is a dense home omorph i c e m b e dding. Pr o o f. Let f b e a dense homeomorphic em b edding of X in Y . Then f ( X ) is a lo cally compact dense subspace of Y and hence it is op en in Y . Th us f is an o p en injection, i.e. f is an InOpLC -morphism. Therefore, b y Theorem 1.12, ϕ is a DInOpLC - morphism. Put Z = f ( X ) and let i : Z − → Y b e the em b edding of Z in Y . Then ψ = Ψ t ( i ) : R C ( Y ) − → R C ( Z ) is defined by the form ula ψ ( F ) = cl Z ( Z ∩ in t Y ( F )) = F ∩ Z , f or ev ery F ∈ RC ( Y ). Hence, by Lemma 2.1, ψ is a Bo olean isomorphism. Since f = i ◦ ( f ↾ X ), w e obtain that ϕ is a Bo olean isomorphism as w ell. Con ve rsely , if ϕ : ( A, ρ, I B) − → ( B , η , I B ′ ) is a DInOpLC - morphism and a Bo olean isomorphism then, by Theorem 1.12, f = Ψ a ( ϕ ) is a homeomorphic em- b edding. Let X = Ψ a ( B , η , I B ′ ), Y = Ψ a ( A, ρ, I B) and ˆ ϕ = Ψ t ( f ). The n Theorem 1.12 show s that ˆ ϕ is a Bo olean isomorphism. Thus , if F ∈ R C ( Y ) and F 6 = ∅ then ˆ ϕ ( F ) 6 = ∅ , i.e. cl( f − 1 (in t( F )) 6 = ∅ . This means that f ( X ) ∩ in t( F ) 6 = ∅ . Therefore, f ( X ) is dense in Y . Prop osition 2.3 L et f ∈ Sk eLC ( X , Y ) and ϕ = Ψ t ( f ) . Then f is a home omo r- phic emb e dding iff ϕ = ϕ 1 ◦ ϕ 2 wher e ϕ 1 is a DInSk eP erLC -m orphism and ϕ 2 is a Bo ole an isomorphism and a DInOpLC -morphism. 7 Pr o o f. Let f : X − → Y b e a homeomorphic embedding. Put Z = cl Y ( f ( X )) and let f 1 : X − → Z b e the restriction of f , f 2 : Z − → Y b e the em b edding of Z in Y . Then f = f 2 ◦ f 1 . Ob viously , f 1 is an op en map. Let U b e an op en subset of Z . Then V = f − 1 ( U ) is op en in X and f ( V ) ⊆ f 2 ( U ). Now, the fact that f is a sk eletal map and [4, Lem ma 2 .4] imply that f 2 is a sk eletal map. Hence, f i , i = 1 , 2 , are Sk eLC -morphisms. Set ϕ i = Ψ t ( f i ), i = 1 , 2. Then ϕ = ϕ 1 ◦ ϕ 2 and Theorem 2.2 together with Theorem 1.10 sho w that ϕ i , i = 1 , 2, are as required. Con ve rsely , let ϕ = ϕ 1 ◦ ϕ 2 , where ϕ 1 is a DInSk eP er LC - morphism and ϕ 2 is a DInOpLC - morphism and a Bo olean isomorphism. Set ˆ f = Ψ a ( ϕ ) and ˆ f i = Ψ a ( ϕ i ), i = 1 , 2. Then ˆ f = ˆ f 2 ◦ ˆ f 1 and, b y Theorems 2.2 and 1 .10, ˆ f 1 is a sk eletal dense ho meomorphic embedding and ˆ f 2 is a sk eletal closed ho meomorphic em b edding. Hence ˆ f is a sk eletal homeomorphic em b edding. Then, b y [4, Th eorem 2.11], f is a sk eletal homeomorphic em b edding. Recall that a contin uous mapping f : X − → Y is said to b e semi-op e n ([17]) if for ev ery p oint y ∈ f ( X ) there exists a p o in t x ∈ f − 1 ( y ) suc h that, for ev ery U ⊆ X , x ∈ in t X ( U ) implies that y ∈ in t f ( X ) ( f ( U )). The following assertion is a sligh t generalization of [7, Theorem 6]. Theorem 2.4 A morphism ϕ ∈ DSk eLC (( A, ρ, I B) , ( B , η , I B ′ )) is an LCA-emb e d d - ing iff the map f = Ψ a ( ϕ ) is a quasi-op en semi-op en p erfe ct surje ction. Pr o o f. Set X = Ψ a ( A, ρ, I B), Y = Ψ a ( B , η , I B ′ ), C = C ρ and C ′ = C η (see [4, 1.16] for the no tations). Then, by [4, 1.16], ( A, C ) and ( B , C ′ ) are CNCA’s. By the pro of of Theorem [4, 2.1], Ψ a ( A, C ) = α X = X ∪ { σ A ∞ } and Ψ a ( B , C ′ ) = αY = Y ∪ { σ B ∞ } . Let ϕ b e an LCA-embedding, i.e. ϕ : A − → B is a Bo olean embedding suc h that, for any a, b ∈ A , aρb iff ϕ ( a ) η ϕ ( b ), and a ∈ I B iff ϕ ( a ) ∈ I B ′ ; hence ϕ satisfies condition (L3) from [4]. Then, b y Theorem 1.8 , f is a p erfect quasi-op en s urjection. It remains to sho w that f is semi-op en. Denote by ϕ c the map ϕ regarded as a function from ( A, C ) to ( B , C ′ ). By [4, (55) ], ϕ c satisfies condition (F1 ) fro m [4, 2.12]. W e will sho w tha t ϕ c is a n NCA-em b edding. Indeed, fo r an y a, b ∈ A , w e ha ve that aC b iff aρb or a, b 6∈ I B; since ϕ is an LCA-em b edding, w e o btain that aC b iff ϕ c ( a ) C ′ ϕ c ( b ). So, ϕ c is an NCA-em b edding and an DSk eC -morphism. Then, b y Theorem 6 of F edorc h uk’s pap er [7], f c = Ψ a ( ϕ c ) : αY − → αX is a semi-op en map. If 1 A 6∈ I B and 1 B 6∈ I B ′ then f − 1 c ( σ A ∞ ) = { σ B ∞ } (see the pro of o f [4 , Theorem 2.15]) and since f = f c | Y , we obtain t hat f is semi-op en. F urther, if 1 A ∈ I B and 1 B ∈ I B ′ then the fact that f is semi-op en is ob vious. Since only these t w o cases are p ossible in the giv en situation, w e ha ve pro v ed tha t f is a p erfect quasi-op en semi-op en surjection. Con ve rsely , let f b e a p erfect quasi-op en semi-op en surjection. Then, b y T he- orem 1.8, ϕ is an injectiv e DSk ePerLC -morphism. Hence ϕ Λ ◦ ϕ = id A . Th us, if ϕ ( a ) ∈ I B ′ then, b y (L2), a = ϕ Λ ( ϕ ( a )) ∈ I B. Using (L3 ), w e obtain that a ∈ I B iff ϕ ( a ) ∈ I B ′ . Since (L1) tak es place, it remains only to pro v e that aρb implies ϕ ( a ) η ϕ ( b ), for all a, b ∈ A . Set ˆ ϕ = Ψ t ( f ). L et F , G ∈ RC ( X ), 8 F ∩ G 6 = ∅ and x ∈ F ∩ G . Set U = int( F ) and V = in t ( G ). Then x ∈ cl( U ) ∩ cl( V ). Since f is a semi-op en surjection, the re exists y ∈ f − 1 ( x ) suc h that, for ev ery W ⊆ Y , y ∈ in t Y ( W ) implies that x ∈ in t X ( f ( W )). W e will sho w that y ∈ cl( f − 1 ( U )) ∩ cl( f − 1 ( V ) ) . Indeed, supp ose that y 6∈ cl( f − 1 ( U )). The n there exists a n op en neighborho o d O y of y suc h that O y ∩ f − 1 ( U ) = ∅ . Th us f ( O y ) ∩ U = ∅ . Since x ∈ cl( U ) and x ∈ in t( f ( O y )), w e obtain a con tradiction. Hence y ∈ cl ( f − 1 ( U )). Analogously w e can sho w that y ∈ cl( f − 1 ( V )). There fore, y ∈ cl( f − 1 ( U )) ∩ cl( f − 1 ( V )) = ˆ ϕ ( F ) ∩ ˆ ϕ ( G ). No w, using [4, Theorem 2.15], w e get that aρb implies ϕ ( a ) η ϕ ( b ). Definition 2.5 Let ( A, ρ, I B) b e a CLCA. An ideal I of A is called a δ -ide a l if I ⊆ I B and fo r any a ∈ I there exists b ∈ I suc h that a ≪ ρ b . If I 1 and I 2 are tw o δ -ideals of ( A, ρ, I B) then we put I 1 ≤ I 2 iff I 1 ⊆ I 2 . W e will denote by ( I ( A, ρ, I B) , ≤ ) the p oset of all δ -ideals o f ( A, ρ, I B). F act 2.6 L et ( A, ρ, I B) b e a CLCA. Then, for every a ∈ A , the set { b ∈ I B | b ≪ ρ a } is a δ -ide al . Such δ -ide als wil l b e c al le d principal δ -ideals . Pr o o f. The pro of is ob vious. Recall that a fr ame is a complete la ttice L satisfying the infinite distributiv e la w a ∧ W S = W { a ∧ s | s ∈ S } , fo r ev ery a ∈ L and ev ery S ⊆ L . F act 2.7 L et ( A, ρ, I B) b e a CLCA. Then the p oset ( I ( A, ρ, I B) , ≤ ) of al l δ -ide als of ( A, ρ, I B) (se e 2.5) is a fr ame. Pr o o f. It is w ell kno wn that the set I dl ( A ) of all ideals of a distributiv e lattice forms a frame under the inclusion ordering (see , e.g., [9 ]). It is easy to see that the join in ( I dl ( A ) , ⊆ ) o f a family of δ -ideals is a δ -ideal and hence it is the join of this family in ( I ( A, ρ, I B) , ≤ ). The meet in ( I dl ( A ) , ⊆ ) of a finite family of δ -ideals is also a δ -ideal and hence it is the meet of this family in ( I ( A, ρ, I B) , ≤ ). Therefore, ( I ( A, ρ, I B) , ≤ ) is a fra me. Note that the meet of an infinite family of δ -ideals in ( I ( A, ρ, I B) , ≤ ) is not obliged to coincide with the meet of the same family in ( I dl ( A ) , ⊆ ). Theorem 2.8 L et ( A, ρ, I B) b e a CLCA, Y = Ψ a ( A, ρ, I B) a n d O ( Y ) b e the fr ame of al l op en subsets of Y . Then ther e exists a fr am e isom o rphism ι : ( I ( A, ρ, I B) , ≤ ) − → ( O ( Y ) , ⊆ ) , wher e ( I ( A, ρ, I B) , ≤ ) is the f r am e of al l δ -id e als of ( A, ρ, I B) . The isomorphism ι sends the set P I ( A, ρ, I B) of al l princ i p a l δ -ide als of ( A, ρ, I B) o n to the set R O ( Y ) of al l r e gular op en subsets of Y . 9 Pr o o f. Let I b e a δ -ideal. Put ι ( I ) = S { λ g A ( a ) | a ∈ I } . Then ι ( I ) is an op en subset of Y . Indeed, for ev ery a ∈ I there exists b ∈ I suc h tha t a ≪ b . Then λ g A ( a ) ⊆ in t Y ( λ g A ( b )) ⊆ λ g A ( b ) ⊆ ι ( I ). Hence ι ( I ) is an op en subset of Y . Therefore ι is a function from I ( A, ρ, I B) to O ( Y ). Let U ∈ O ( Y ). Set I B U = { b ∈ I B | λ g A ( b ) ⊆ U } . Then, as it is easy to see, I B U is a δ -ideal of ( A, ρ, I B) . Since Y is a lo cally compact Hausdorff space, ι (I B U ) = U . Hence, ι is a surjection. W e will sho w that ι is a n injection as w ell. Indee d, let I 1 , I 2 ∈ I ( A, ρ, I B) and ι ( I 1 ) = ι ( I 2 ). Set ι ( I 1 ) = W and put I B W = { b ∈ I B | λ g A ( b ) ⊆ W } . Then, obviously , I 1 ⊆ I B W . F urther, if b ∈ I B W then λ g A ( b ) ⊆ W and λ g A ( b ) is compact. Since I 1 is a δ -ideal, Ω = { in t( λ g A ( a )) | a ∈ I 1 } is an op en co v er of W and, hence, of λ g A ( b ). Th us there exists a finite sub cov er { λ g A ( a 1 ) , . . . , λ g A ( a k ) } o f Ω. Therefore, λ g A ( b ) ⊆ S { λ g A ( a i ) | i = 1 , . . . , k } = λ g A ( W { a i | i = 1 , . . . , k } ). This implies that b ≤ W { a i | i = 1 , . . . , k } and hence b ∈ I 1 . So , w e ha v e pro v ed tha t I 1 = I B W . Analogously we can show that I 2 = I B W . Th us I 1 = I 2 . Therefore, ι is a bijection. It is ob vious t hat if I 1 , I 2 ∈ I ( A, ρ, I B) and I 1 ≤ I 2 then ι ( I 1 ) ⊆ ι ( I 2 ). Con v ersely , if ι ( I 1 ) ⊆ ι ( I 2 ) then I 1 ≤ I 2 . Indeed, if ι ( I i ) = W i , i = 1 , 2, then, as w e ha ve already seen, I i = I B W i , i = 1 , 2; since W 1 ⊆ W 2 implies that I B W 1 ⊆ I B W 2 , w e get that I 1 ≤ I 2 . So, ι : ( I ( A, ρ, I B) , ≤ ) − → ( O ( Y ) , ⊆ ) is an isomorphism of p osets. This implies that ι is also a frame isomorphism. Let U b e a regular op en subset of Y . Set F = Y \ U . Then there exists a ∈ A suc h that F = λ g A ( a ). Put I B U = { b ∈ I B | λ g A ( b ) ⊆ U } . Then, as w e hav e already seen, I B U is a δ -ideal and ι (I B U ) = U . Since F ∈ RC ( Y ), we ha v e tha t I B U = { b ∈ I B | b ≪ ρ a ∗ } . Hence I B U is a principal δ -ideal. Con ve rsely , if I is a principal δ - ideal then U = ι ( I ) is a regular op en set in Y . Indeed, let a ∈ A and I = { b ∈ I B | b ≪ ρ a } . It is enough t o prov e that Y \ U = λ g A ( a ∗ ). If b ∈ I then b ( − ρ ) a ∗ and hence λ g A ( b ) ∩ λ g A ( a ∗ ) = ∅ . Th us U ⊆ Y \ λ g A ( a ∗ ). If σ ∈ Y \ λ g A ( a ∗ ) then, by [4, (24)], there exists b ∈ I B suc h that σ ∈ λ g A ( b ) ⊆ Y \ λ g A ( a ∗ ). Since, b y [4, (11)] and [4, (42)], Y \ λ g A ( a ∗ ) = int Y ( λ g A ( a )), w e get that b ≪ ρ a . Therefore b ∈ I and hence σ ∈ U . This means that Y \ λ g A ( a ∗ ) ⊆ U . Recall that if A is a Bo olean a lgebra and a ∈ A then the set ↓ a endo w ed with the same meets and joins as in A and with complemen ts ¬ b defined b y the form ula ¬ b = b ∗ ∧ a , for ev ery b ≤ a , is a Bo olean algebra; it is denoted b y A | a . If J = ↓ a ∗ then A | a is isomorphic to the fa ctor algebra A/J ; the isomorphism h : A | a − → A/J is the following: h ( b ) = [ b ], for ev ery b ≤ a (see, e.g., [14]). With the next the orem w e sho w ho w one can build the C LCAs corresponding to the open subsets of a lo cally compact Hausd orff space Y from the CLCA Ψ t ( Y ). Theorem 2.9 L et ( A, ρ, I B) b e a CLCA, Y = Ψ a ( A, ρ, I B) and I b e a δ -ide al of ( A, ρ, I B) . Set a I = W I and B = A | a I . F or every a, b ∈ B , set aη b iff ther e exi s t c ∈ I and σ ∈ Y such that a, b, c ∈ σ . T hen ( B , η , I ) is a CLCA. L et ϕ : A − → B b e the natur al epimorp h ism (i.e. ϕ ( a ) = a ∧ a I , for every a ∈ A ). If X = Ψ a ( B , η , I ) and f = Ψ a ( ϕ ) then f : X − → Y is an op en inje ction and f ( X ) = ι ( I ) (se e 2.8 for 10 ι ). (Henc e, X is home o m orphic to ι ( I ) .) Pr o o f. Ob viously , B is a complete Bo olean a lg ebra and ϕ is a surjectiv e complete Bo olean homomorphism. Set U = ι ( I ) (i.e. U = S { λ g A ( b ) | b ∈ I } ). Then U is op en in Y (see 2.8) and cl( U ) = λ g A ( a I ). Since I is a δ -ideal, { in t( λ g A ( b )) | b ∈ I } is an op en cov er of U . If I = { 0 } then U = ∅ , a I = 0, B = { 0 } and X = ∅ ; hence, in this case the assertion of the theorem is true. Thu s, let us a ssume that I 6 = { 0 } . W e will first c heck that ( B , η , I ) is a CLCA, i.e. that conditions (C1)-(C4) and (BC1)-(BC3) from [4] are fulfilled. Let b ∈ B \ { 0 } . Set V = int( λ g A ( b )). Since b ≤ a I , w e obt a in that ∅ 6 = V ⊆ λ g A ( b ) ⊆ λ g A ( a I ) = cl ( U ) . Hence V ∩ U 6 = ∅ . Since Y is lo cally compact, there exists c ∈ I B \ { 0 } suc h t ha t λ g A ( c ) ⊆ V ∩ U . Th us λ g A ( c ) ⊆ U and, hence, there exists d ∈ I such tha t λ g A ( c ) ⊆ λ g A ( d ) (w e use the fact that λ g A ( c ) is compact and I is closed under finite joins). W e get t ha t c ≤ d . T herefore, c ∈ I . There exists σ ∈ Y suc h that c ∈ σ . Since c ≤ b , w e obtain that b ∈ σ . Hence, bη b . So, the axiom (C1) is fulfilled. Note that c ≪ ρ b (b ecause λ g A ( c ) ⊆ V = int( λ g A ( b ))). Thus , c ≪ η b . Therefore, the axiom (BC3) is chec ked as well. Clearly , the axioms (C2) and (C3) are satisfied. Using condition (K2) from [4], w e obtain that the axiom (C4) is also fulfilled. Let a ∈ I , c ∈ B and a ≪ η c . Then, for ev ery σ ∈ Y , we hav e t hat either a 6∈ σ or ( c ∗ ∧ a I ) 6∈ σ . Since a ∈ I B, we get that a ( − C ρ )( c ∗ ∧ a I ) (see [1 9, Lemma 20]). Using ag ain the fact that a ∈ I B, w e obtain that a ≪ ρ ( c ∨ a ∗ I ). Then there exists b ∈ I B such that a ≪ ρ b ≪ ρ ( c ∨ a ∗ I ). Since I is a δ - ideal, there exists d ∈ I suc h that a ≪ ρ d . Then a ≪ ρ b ∧ d and b ∧ d ∈ I . Thus a ≪ η b ∧ d ≪ η c . Therefore, the axiom (BC1) is c hec k ed. Let us note that if σ is a cluster in ( A, C ρ ) , a ∈ σ and b ∗ 6∈ σ then a ∧ b ∈ σ . (1) Indeed, there exists an ultrafilter u suc h that a ∈ u ⊆ σ . Then b ∗ 6∈ u and hence b ∈ u . Therefore a ∧ b ∈ u . Th us a ∧ b ∈ σ . So, (1) is prov ed. Let a, b ∈ B and aη b . Then there exist c ∈ I and σ ∈ Y suc h that a, b, c ∈ σ . Since I is a δ - ideal, there exists d ∈ I suc h that c ≪ ρ d . Then d ∗ 6∈ σ . Therefore, b y (1 ), b ∧ d ∈ σ . Since a, b ∧ d ∈ σ , w e obtain that aη ( b ∧ d ). Hence, the axiom (BC2) is fulfilled. So, w e ha v e pro ved that ( B , η , I B ′ ) is a CLCA. W e will sho w that ϕ satisfies axioms (L 1 ), (L2) and (LO) f rom [4]. Not e first that, for ev ery a ∈ B , ϕ Λ ( a ) = a. (2) This observ ation sho ws that ϕ satisfies condition (L 2 ). F or c hec king condition ( EL1) from [4] (whic h is equiv alent to the condition (L1) ) , let a, b ∈ B and aη b . Then there exist c ∈ I a nd σ ∈ Y suc h tha t a, b, c ∈ σ . Since I is a δ -ideal, t here exists d ∈ I 11 suc h that c ≪ ρ d . Then d ∗ 6∈ σ and (1 ) implies tha t a ∧ d, b ∧ d ∈ σ . Since a ∧ d, b ∧ d ∈ I B, w e get that ( a ∧ d ) ρ ( b ∧ d ). Therefore, aρb . So, ϕ satisfies condition (EL1). Let us pro v e t ha t the axiom (LO) is fulfilled as w ell. Let a ∈ A , b ∈ I and aρb . Then aC ρ b and since b ∈ I B, Lemma 20 from [19] implies that there exists σ ∈ Y suc h t hat a, b ∈ σ . There exists d ∈ I suc h that b ≪ ρ d (b ecause I is a δ - ideal). Then b ≪ C ρ a I . Thus a ∗ I 6∈ σ . Now , (1) implies that a ∧ a I ∈ σ . Hence ϕ ( a ) η b . Therefore, condition (LO) is che c ke d. So, ϕ satisfies axioms (L 1 ), (L2) and (LO). Then, b y Theorem 1.12, f : X − → Y is a n op en injection and hence f is a homeomorphism b etw een X a nd f ( X ). Let us show that f ( X ) = U . Set ˆ ϕ = Ψ t ( f ). Then ˆ ϕ : R C ( Y ) − → RC ( X ), ˆ ϕ ( F ) = f − 1 ( F ) for ev ery F ∈ R C ( Y ) and λ g B ◦ ϕ = ˆ ϕ ◦ λ g A (see [4, Theorem 2.17]). Hence, for ev ery b ∈ I , λ g B ( b ) = f − 1 ( λ g A ( b )) ⊆ f − 1 ( U ). Since X = S { λ g B ( b ) | b ∈ I } , w e obtain that f ( X ) ⊆ U . F or sho wing that f ( X ) ⊇ U , let σ ∈ U . Then, b y the definition of U , there exists b ∈ I suc h that σ ∈ λ g A ( b ) ⊆ U . Hence b ∈ σ and a ∗ I 6∈ σ . There exists u ∈ U lt ( A ) suc h that b ∈ u ⊆ σ a nd σ = σ u . Since a ∗ I 6∈ σ , w e obtain that a I ∈ u . Th us, for ev ery c ∈ u , c ∧ a I 6 = 0. Therefore, ϕ ( c ) 6 = 0, for eve ry c ∈ u . This implies that ϕ ( u ) is a basis of a b o unded filter in B . Then there exists v ∈ U l t ( B ) suc h that ϕ ( u ) ⊆ v . Hence u ⊆ ϕ − 1 ( v ) and th us u = ϕ − 1 ( v ). Since σ v is a b ounded cluster, w e get that σ v ∈ X . F urther, f ( σ v ) = σ ϕ − 1 ( v ) = σ u = σ . Hence f ( X ) = U . W e will now s ho w ho w one can build the CLCAs correspo nding to the regular closed subsets of a lo cally compact Hausdorff space Y f rom the CLCA Ψ t ( Y ). Theorem 2.10 L et ( A, ρ, I B) b e a CLCA, Y = Ψ a ( A, ρ, I B) , a 0 ∈ A an d F = λ g A ( a 0 ) . Set B = A | a 0 and let ϕ : A − → B b e the natur al epimorphism (i.e. ϕ ( a ) = a ∧ a 0 , for every a ∈ A ). Put I B ′ = ϕ (I B) and let, for every a, b ∈ B , aη b iff aρb . Then ( B , η , I B ′ ) is a CLCA. If X = Ψ a ( B , η , I B ′ ) and f = Ψ a ( ϕ ) then f : X − → Y is a close d quasi-op en inj e c tion and f ( X ) = F . (Henc e, X is home omorphic to F .) Pr o o f. W e ha ve that B is a complete Bo olean algebra, ϕ is a complete Bo olean homomorphism and ϕ Λ ( a ) = a , fo r ev ery a ∈ B . Set ψ = λ g A ◦ ϕ Λ . W e will show that ψ ′ = ψ ↾ B is a CLCA-isomorphism b et w een ( B , η , I B ′ ) and ( R C ( F ) , ρ F , C R ( F )). Since F ∈ RC ( Y ), w e hav e, as it is w ell kno wn, that R C ( F ) ⊆ RC ( Y ) and R C ( F ) = { G ∧ F | G ∈ RC ( Y ) } ; moreo v er, RC ( F ) = RC ( Y ) | F . Hence ψ ′ : B − → R C ( F ) is a Bo olean isomorphism. F or any a, b ∈ B , w e hav e that aη b ⇐ ⇒ ϕ Λ ( a ) ρϕ Λ ( b ) ⇐ ⇒ λ g A ( ϕ Λ ( a )) ∩ λ g A ( ϕ Λ ( b )) 6 = ∅ ⇐ ⇒ ψ ( a ) ρ F ψ ( b ). Finally , f o r any a ∈ B , w e hav e that a ∈ I B ′ ⇐ ⇒ ϕ Λ ( a ) ∈ I B ⇐ ⇒ λ g A ( ϕ Λ ( a )) is compact ⇐ ⇒ ψ ( a ) ∈ C R ( F ). Therefore, ( B , η , I B ′ ) is a CLCA; it is isomorphic to the CLCA ( RC ( F ) , ρ F , C R ( F )). F or sho wing tha t f : X − → Y is a homeomorphic embedding and f ( X ) = F , note that ϕ satisfies conditions (L1 )-(L3) from [4 ] and condition ( L S), and hence, b y Theorem 1.10, f is a quasi-op en p erfect injection, i.e. f is a homeomorphic em b edding. F rom [4, (45)] w e get that, for ev ery b ∈ I B ′ , f ( λ g B ( b )) = λ g A ( ϕ Λ ( b )) = λ g A ( b ) ⊆ F . Since X = S { λ g B ( b ) | b ∈ I B ′ } , w e o btain that f ( X ) ⊆ F . Let 12 y ∈ in t Y ( F ). Then there exists b ∈ I B suc h that y ∈ in t ( λ g A ( b )) ⊆ λ g A ( b ) ⊆ in t Y ( F ). Hence b ∈ I B ′ . Using again [4, (4 5)], w e get that y ∈ f ( λ g B ( b )), i.e. y ∈ f ( X ). Th us in t Y ( F ) ⊆ f ( X ). Since f ( X ) is closed in Y , w e conclude that f ( X ) ⊇ F . Therefore, f ( X ) = F . 3 Some applications W e start with a prop osition which has a straigh tforw a rd pro o f. Prop osition 3.1 L et ( A, ρ, I B) b e a CLCA, Y = Ψ a ( A, ρ, I B) and a ∈ A . Then a is an atom of A iff λ g A ( a ) is an isolate d p oint of the sp ac e Y . Corollary 3.2 L et ( A, ρ, I B) b e a CLCA. Then I B c ontain s al l finite sums o f the atoms of A . Prop osition 3.3 L et ( A, ρ, I B) b e a CLCA and Y = Ψ a ( A, ρ, I B) . Then Y is a discr ete sp a c e iff I B is the se t of al l fin i te s ums of the a tom s of A . Pr o o f. It follows easily from 3.1 and the fact that Y = S { λ g A ( a ) | a ∈ I B } . The next prop o sition has an easy pro of whic h will b e omitted. Prop osition 3.4 L et ( A, ρ, I B) b e a C LCA and Y = Ψ a ( A, ρ, I B) . Then Y is an extr e mal ly d isc onne cte d sp ac e iff a ≪ ρ a , for every a ∈ A . Prop osition 3.5 L et ( A, ρ, I B) b e a CLCA an d Y = Ψ a ( A, ρ, I B) . Then the set of al l isolate d p oints of Y is dense in Y iff A is an atomic Bo ole an algebr a. Pr o o f. ( ⇒ ) Let X b e the set of all isolated p oints of Y and let cl Y ( X ) = Y . Then, b y 2.1, the Bo o lean algebras RC ( X ) and RC ( Y ) are isomorphic. F rom [4 , Theorem 2.11] w e get that A is isomorphic to RC ( Y ) and, hence, t o RC ( X ). Therefore, A is an atomic Bo o lean algebra. ( ⇐ ) Set X = { λ g A ( a ) | a ∈ Atoms ( A ) } . Then, b y Prop osition 3.1, X consists of isolated p oin ts of Y . W e need only t o sho w that X is dense in Y . Let b ∈ I B and b 6 = 0. Then there exists a ∈ Atoms ( A ) suc h that a ≤ b . Th us λ g A ( a ) ∈ λ g A ( b ), i.e. X ∩ λ g A ( b ) 6 = ∅ . Since Y is a regular space, [4 , (24)] implies that X is dense in Y . Notation 3.6 Let X b e a T yc honoff space. W e will denote b y L ( X ) the se t of all, up to equiv a lence, lo cally compact Hausdorff extensions of X (recall that tw o (lo cally compact Hausdorff ) extens ions ( Y 1 , f 1 ) and ( Y 2 , f 2 ) of X are said to b e e quivalent iff there exists a homeomorphism h : Y 1 − → Y 2 suc h that h ◦ f 1 = f 2 ). W e will regard t wo orders on it. Let [( Y i , f i )] ∈ L ( X ), where i = 1 , 2. W e set [( Y 1 , f 1 )] ≤ [( Y 2 , f 2 )] (resp ectiv ely , [( Y 1 , f 1 )] ≤ s [( Y 2 , f 2 )]) iff there exists a contin uous (resp., con tinuous surjectiv e) mapping h : Y 2 − → Y 1 suc h that f 1 = h ◦ f 2 . 13 Definition 3.7 Let X b e a lo cally compact Hausdorff space a nd Ψ t ( X ) = ( A, ρ, I B). W e will denote b y L a ( X ) the set of all LCAs of the form ( A, ρ 1 , I B 1 ) whic h satisfy the follow ing conditions: (LA1) ρ ⊆ ρ 1 ; (LA2) I B ⊆ I B 1 ; (LA3) for ev ery a ∈ A and ev ery b ∈ I B, bρ 1 a implies bρa . W e will define t w o o rders on the set L a ( X ). If ( A, ρ i , I B i ) ∈ L a ( X ), where i = 1 , 2, w e set ( A, ρ 1 , I B 1 )  ( A, ρ 2 , I B 2 ) (respectiv ely , ( A, ρ 1 , I B 1 )  s ( A, ρ 2 , I B 2 )) iff ρ 2 ⊆ ρ 1 and I B 2 ⊆ I B 1 (and, resp ectiv ely , in addition, fo r ev ery b ounded ultrafilter u in ( A, ρ 1 , I B 1 ) there exists b ∈ I B 2 suc h that bρ 1 u ). Theorem 3.8 L et ( X, τ ) b e a lo c al ly c omp act Hausdorff sp ac e and le t Ψ t ( X , τ ) = ( A, ρ, I B) . Then ther e exists an isomorphism µ (r es p e c tively, µ s ) b e twe e n the or der e d sets ( L ( X ) , ≤ ) and ( L a ( X ) ,  ) (r esp e ctively, ( L ( X ) , ≤ s ) and ( L a ( X ) ,  s ) ). Pr o o f. If X is compact then ev erything is clear. Th us, let X b e a non-compact space. Let [( Y , f )] ∈ L ( X ). Then f : X − → Y is a dense homeomorphic em b edding and hence it is an op en injection . Set ( A ′ , ρ ′ , I B ′ ) = Ψ t ( Y ) and ϕ = Ψ t ( f ). Then , b y Theorem 2.2, ϕ : ( A ′ , ρ ′ , I B ′ ) − → ( A, ρ, I B) is a Bo olean isomorphism and a DInOpLC -morphism. F or ev ery a, b ∈ A , set aρ f b iff ϕ − 1 ( a ) ρ ′ ϕ − 1 ( b ), and set a ∈ I B f iff ϕ − 1 ( a ) ∈ I B ′ . F or eve ry a ∈ A , set ψ ( a ) = ϕ − 1 ( a ). Then ψ : ( A, ρ f , I B f ) − → ( A ′ , ρ ′ , I B ′ ) is an LCA-isomorphism. It is easy to see t ha t ( A, ρ f , I B f ) ∈ L a ( X ). Set µ ([( Y , f )]) = ( A, ρ f , I B f ). Then it is not difficult to sho w that µ is a w ell-defined order preserving map b et wee n the o rdered sets ( L ( X ) , ≤ ) and ( L a ( X ) ,  ). Let ( A, ρ 1 , I B 1 ) ∈ L a ( X ). Then the iden tity map i : ( A, ρ 1 , I B 1 ) − → ( A, ρ, I B) is a Bo olean isomorphism and a DInOpLC -morphism. Set X ′ = Ψ a ( A, ρ, I B), Y = Ψ a ( A, ρ 1 , I B 1 ) and f ′ = Ψ a ( i ). Then, by Theorem 2.2, f ′ : X ′ − → Y is a dense homeomorphic em b edding. Set f = f ′ ◦ t X (see [4, (26)] for the definition of the homeomorphism t X : X − → X ′ ). Then [( Y , f )] ∈ L ( X ) and w e put µ ′ ( A, ρ 1 , I B 1 ) = [( Y , f )]. It is easy to show that µ ′ : ( L a ( X ) ,  ) − → ( L ( X ) , ≤ ) is an order preserving map. Using [4, Theorem 2.11], it is not difficult to pro v e that the comp o sitions µ ◦ µ ′ and µ ′ ◦ µ a re iden tit ies. Finally , we will sho w that the same map µ , whic h will b e no w denoted b y µ s , is an isomorphism b et w een the ordered sets ( L ( X ) , ≤ s ) and ( L a ( X ) ,  s ). This can b e prov ed easily using Theorem 1.3 and [5, Prop osition 3.2] (the last prop osition sa ys that if ( B , η ) is a CA and F 1 , F 2 are t wo filters in B suc h that F 1 η F 2 then there exist ultrafilters u 1 , u 2 in B suc h that F i ⊆ u i , where i = 1 , 2, and u 1 η u 2 ). This completes the pro of of our theorem. Recall that if X is a set and P ( X ) is the p o we r set of X o rdered by the inclusion, then a triple ( X , ρ, I B) is called a lo c al pr oxim ity sp ac e (see [10]) if ( P ( X ) , ρ ) is a CA, I B is an ideal ( p ossibly non prop er) of P ( X ) and the axioms ( BC1),( BC2) from [4, 14 Definition 1.14 ] are f ulfilled. A lo cal proximit y space ( X , ρ, I B) is said to b e sep ar ate d if ρ is the iden tit y relatio n on singletons. Remark 3.9 In this remark we will use the notatio ns fr o m Theorem 3.8. Let ( A, ρ 1 , I B 1 ) ∈ L a ( X ) and µ − 1 ( A, ρ 1 , I B 1 ) = [( Y , f ) ]( ∈ L ( X )). Let ( X , ρ Y , I B Y ) b e the lo cal proximit y space induced b y ( Y , f ) (i.e., for ev ery M , N ⊆ X , M ρ Y N ⇐ ⇒ cl Y ( f ( M ) ) ∩ cl Y ( f ( N )) 6 = ∅ and M ∈ I B Y iff cl Y ( f ( M ) ) is compact). Then I B 1 = I B Y ∩ RC ( X ) and, fo r ev ery F , G ∈ RC ( X ), F ρ Y G ⇐ ⇒ F ρ 1 G . Indeed, using [4, Theorem 2.11 and (31)] and the fact that f ′ is an op en ma pping, w e obtain that, for ev ery F ∈ RC ( X ), λ g ( A,ρ, I B) ( F ) = ( f ′ ) − 1 ( λ g ( A,ρ 1 , I B 1 ) ( F )) and thus, by 2 .1, cl Y ( f ′ ( λ g ( A,ρ, I B) ( F ))) = cl Y ( f ′ (( f ′ ) − 1 ( λ g ( A,ρ 1 , I B 1 ) ( F )))) = λ g ( A,ρ 1 , I B 1 ) ( F ). As it f ollo ws from [4, Theorem 2.11], Ψ t ( t X ) ◦ λ g ( A,ρ, I B) = id A . Hence, using [4, (31)], w e obtain that, for ev ery F ∈ RC ( X ), F = (Ψ t ( t X ))( λ g ( A,ρ, I B) ( F )) = cl X ′ (in t X ′ ( t − 1 X ( λ g ( A,ρ, I B) ( F )))) and th us F = t − 1 X ( λ g ( A,ρ, I B) ( F )), i.e. t X ( F ) = λ g ( A,ρ, I B) ( F ). Therefore, cl Y ( f ( F )) = λ g ( A,ρ 1 , I B 1 ) ( F ) for ev ery F ∈ RC ( X ). No w, all follows from [4, (I I I) and (IV) in the pro of of Theorem 2.1]. Notation 3.10 If ( A, ρ, I B) is a CLCA then w e will write ρ ⊆ I B C prov ided that C is a normal contact relation on A satisfying the follo wing conditions: (R C1) ρ ⊆ C , and (R C2) for ev ery a ∈ A and ev ery b ∈ I B, aC b implies aρb . If ρ ⊆ I B C 1 and ρ ⊆ I B C 2 then w e will write C 1  c C 2 iff C 2 ⊆ C 1 . Corollary 3.11 L et ( X , τ ) b e a lo c al ly c omp a ct Hausdorff sp ac e and let Ψ t ( X , τ ) = ( A, ρ, I B) . Then ther e exis ts a n iso m orphism µ c b etwe en the o r der e d set ( K ( X ) , ≤ ) of al l, up to e quivalenc e, Hausdorff c omp actific ation s of ( X, τ ) and the or der e d set ( K a ( X ) ,  c ) of al l normal c ontact r elations C on A such that ρ ⊆ I B C (se e 3.10 for the notations). Pr o o f. It follows immediately from Theorem 3.8. Prop osition 3.12 L et ( X, τ ) b e a lo c a l ly c omp act non-c omp act Hausdorff sp ac e and let Ψ t ( X , τ ) = ( A, ρ, I B) . Then C ρ is the smal lest element of the or der e d set ( K a ( X ) ,  c ) (se e [4, L emm a 1.16 ] for C ρ ); henc e, if ( αX , α ) is the Alexandr off (one-p oint) c omp actific ation of X then µ c ([( αX , α )]) = C ρ (se e Cor ol lary 3.11 for µ c ). F urther, the or der e d set ( K a ( X ) ,  c ) has a gr e a test element C β ρ ; it is define d as fol lows : for eve ry a, b ∈ A , a ( − C β ρ ) b iff ther e exists a set { c d ∈ A | d ∈ I D } such that: (1) a ≪ ρ c d ≪ ρ b ∗ , for al l d ∈ I D , a n d (2) for any two elements d 1 , d 2 of I D , d 1 < d 2 implies that c d 1 ≪ ρ c d 2 . Henc e, if ( β X, β ) is the Stone - ˇ Ce ch c om p actific ation of X then µ c ([( β X , β )]) = C β ρ . 15 Pr o o f. Recall that, f o r ev ery a, b ∈ A , aC ρ b iff either aρb or a, b 6∈ I B. Obv iously , C ρ ∈ K a ( X ). It is easy to see that if C ∈ K a ( X ) then C ⊆ C ρ . Hence, b y Corollary 3.11, µ c ([( αX , α )]) = C ρ . The pro of that C β ρ ∈ K a ( X ) is straightforw ard. Let C ∈ K a ( X ). W e will sho w tha t C β ρ ⊆ C . Let a, b ∈ A and aC β ρ b . Supp ose tha t a ( − C ) b . Then a ≪ C b ∗ . Hence there exists c 1 2 ∈ A suc h that a ≪ C c 1 2 ≪ C b ∗ . Since ρ ⊆ C , w e obtain that a ≪ ρ c 1 2 ≪ ρ b ∗ . It is clear no w that w e can construct a set { c d ∈ A | d ∈ I D } suc h that a ≪ ρ c d ≪ ρ b ∗ , for all d ∈ I D, a nd for an y tw o elemen ts d 1 , d 2 of I D, d 1 < d 2 implies that c d 1 ≪ ρ c d 2 . Th us, a ( − C β ρ ) b , a con tradiction. Therefore, aC b . Now , Corollary 3.11 implies that µ c ([( β X , β )]) = C β ρ . Remark 3.13 The definition of the relation C β ρ in Prop osition 3.1 2 is giv en in the language of contact relations. It is clear that if we use the fa ct that all happ ens in a top ological space X then w e can define the relation C β ρ b y setting fo r ev ery a, b ∈ A , a ( − C β ρ ) b iff a and b are completely separated. Prop osition 3.14 L et X b e a lo c al ly c omp act non - c omp act Hausdorff sp ac e. L et Ψ t ( X , τ ) = ( A, ρ, I B) and let { C m | m ∈ M } b e a subset of K a ( X ) (s e e 3.11 for K a ( X ) ). F or every a, b ∈ A , put a ( − C ) b iff ther e exists a set { c d ∈ A | d ∈ I D } such that: (1) a ≪ C m c d ≪ C m b ∗ , for al l d ∈ I D and fo r e ach m ∈ M , and (2) f o r any two elements d 1 , d 2 of I D , d 1 < d 2 implies that c d 1 ≪ C m c d 2 , for every m ∈ M . Then C is the s upr e m um in ( K a ( X ) ,  c ) of the set { C m | m ∈ M } . Pr o o f. The pro of is straigh tforw ar d. Remark 3.15 Note that if X is an infinite discrete space and Ψ t ( X ) = ( A, ρ, I B) then A is t he p ow er set of X ordered by the inclusion, I B is the set of all finite subsets of X and ρ is the smallest no r ma l con tact relation on A . Hence, in this case, ρ = C β ρ . Note also that Prop ositions 3.1 4 and 3.12 imply that ( K a ( X ) ,  c ) is a complete lattice. Now , using Corollary 3.11, w e obtain a new pro of o f the w ell-know n fact that, for ev ery lo cally compact space X , ( K ( X ) , ≤ ) is a complete lattice. W e are now going to strengthen the Leader Lo cal Compactification Theorem ([10]) in the same manner a s de V ries ([2]) strengthen Smirnov Compactification Theorem ([15]). Recall that ev ery lo cal pro ximit y space ( X , ρ, I B) induces a completely regular top ology τ ( X,ρ, I B) in X by defining cl( M ) = { x ∈ X | xρM } for ev ery M ⊆ X ([1 0]). If ( X , τ ) is a top ological space then w e sa y that ( X , ρ, I B) is a lo c al pr oximity sp ac e on ( X , τ ) if τ ( X,ρ, I B) = τ . 16 Lemma 3.16 L et ( X , ρ i , I B i ) , i = 1 , 2 , b e two sep a r a te d l o c a l pr oximity sp ac es on a T ychonoff sp a c e ( X , τ ) such that I B 1 ∩ RC ( X ) = I B 2 ∩ RC ( X ) and ρ 1 | RC ( X ) = ρ 2 | RC ( X ) (i.e., for every F , G ∈ RC ( X ) , F ρ 1 G ⇐ ⇒ F ρ 2 G ). Then ρ 1 = ρ 2 and I B 1 = I B 2 . Pr o o f. L et ( Y i , f i ) b e the lo cally compact extension of X generated b y ( X , ρ i , I B i ), where i = 1 , 2 (see [10]). Let B ∈ I B 1 . Then cl Y 1 ( f 1 ( B )) is compact. There exists an op en subset U of Y 1 suc h that cl Y 1 ( f 1 ( B )) ⊆ U and cl Y 1 ( U ) is compact. Let F = f − 1 1 (cl Y 1 ( U )). Then F ∈ RC ( X ) and cl Y 1 ( f 1 ( F )) = cl Y 1 ( U ). Hence F ∈ I B 1 ∩ RC ( X ). Th us F ∈ I B 2 . Since B ⊆ F , we get that B ∈ I B 2 . Therefore, I B 1 ⊆ I B 2 . Analogously w e obta in that I B 2 ⊆ I B 1 . Thus I B 1 = I B 2 . Let M , N ⊆ X and M ( − ρ 1 ) N . Supp o se that M ρ 2 N . The n there exist M ′ , N ′ ∈ I B 2 suc h that M ′ ⊆ M , N ′ ⊆ N and M ′ ρ 2 N ′ . Since I B 1 = I B 2 , w e get that M ′ , N ′ ∈ I B 1 . Hence K 1 = cl Y 1 ( f 1 ( M ′ )) and K 2 = cl Y 1 ( f 1 ( N ′ )) are disjoint compact subsets of Y 1 . Then there exist op en subsets U and V of Y 1 ha ving dis- join t closures in Y 1 and containing, resp ectiv ely , K 1 and K 2 . Set F = f − 1 1 (cl Y 1 ( U )) and G = f − 1 1 (cl Y 1 ( V ) ) . Then F , G ∈ RC ( X ), M ′ ⊆ F , N ′ ⊆ G a nd F ( − ρ 1 ) G . Th us F ( − ρ 2 ) G and hence M ′ ( − ρ 2 ) N ′ , a con tradiction. Therefore, M ( − ρ 2 ) N . So, ρ 2 ⊆ ρ 1 . Using t he symmetry , w e obtain that ρ 1 = ρ 2 . Definition 3.17 Let ( X , τ ) b e a T yc honoff space. An LCA ( R C ( X, τ ) , ρ, I B) is said to b e admis s ible for ( X, τ ) if it satisfies the following conditions: (A1) if F , G ∈ RC ( X ) and F ∩ G 6 = ∅ t hen F ρG (i.e. ρ X ⊆ ρ ( see [4, 1.10] for ρ X )); (A2) if F ∈ RC ( X ) and x ∈ in t X ( F ) then there exists G ∈ I B suc h that x ∈ in t X ( G ) and G ≪ ρ F . The set of all LCAs whic h ar e admissible f or ( X , τ ) will b e denoted by L ad ( X , τ ) (or simply b y L ad ( X )). If ( R C ( X ) , ρ i , I B i ) ∈ L ad ( X ), where i = 1 , 2, then we set ( RC ( X ) , ρ 1 , I B 1 )  l ( RC ( X ) , ρ 2 , I B 2 ) iff ρ 2 ⊆ ρ 1 and I B 2 ⊆ I B 1 . Lemma 3.18 L et ( X , ρ, I B) b e a sep ar ate d lo c al pr oximity sp ac e. Se t τ = τ ( X,ρ, I B) . L e t ρ ′ = ρ | RC ( X,τ ) and I B ′ = I B ∩ RC ( X , τ ) . Then ( RC ( X, τ ) , ρ ′ , I B ′ ) ∈ L ad ( X , τ ) . Pr o o f. The fa ct that ( R C ( X, τ ) , ρ ′ , I B ′ ) is an LCA is prov ed in [19, Example 40]. The rest can b e easily c hec k ed. Lemma 3.19 L et ( X , τ ) b e a T ychon o ff sp ac e and ( R C ( X ) , ρ ′ , I B ′ ) ∈ L ad ( X ) . T hen ther e exists a unique sep ar ate d lo c al pr oximity sp ac e ( X , ρ, I B) on ( X , τ ) such that I B ′ = RC ( X ) ∩ I B and ρ | RC ( X ) = ρ ′ . Mor e over, I B = { M ⊆ X | ∃ B ∈ I B ′ such that M ⊆ B } and for ev e ry M , N ⊆ X , N ( − ρ ) M ⇐ ⇒ M ( − ρ ) N ⇐ ⇒ ∀ B ∈ I B ∃ F ∈ I B ′ and ∃ G ∈ R C ( X ) s uch that M ∩ B ⊆ in t X ( F ) , N ⊆ in t X ( G ) and F ( − ρ ′ ) G . Pr o o f. The pro of that ( X, ρ, I B) is a separated lo cal pro ximity space on ( X , τ ) is straigh tforw ar d. The uniquenes s follo ws from Lemma 3.16. 17 Theorem 3.20 L et ( X , τ ) b e a T ychono ff sp ac e. Then the or der e d sets ( L ( X , τ ) , ≤ ) and ( L ad ( X , τ ) ,  l ) ar e isomorphic . Pr o o f. It follows fro m Lemmas 3.18 and 3.1 9 and the Leader Lo cal Compactification Theorem ([10]). Remark 3.21 The preceding theorem strengthen Leader Lo cal Compactification Theorem ([10]) b ecause the relation ρ is no w giv en only on the subset R C ( X ) of the p ow er set of X and the same is true fo r the b oundedness I B. Using Remark 3.9, w e see also tha t it is a generalization of t he first part of Theorem 3.8 (i.e. that one concerning t he ordered set ( L ( X ) , ≤ )). In the Leader’s pap er [10] there is no ana lo gue of the second part of the Theorem 3.8 ( i.e. tha t one concerning the ordered set ( L ( X ) , ≤ s )). As far as w e kno w, another description of the ordered set ( L ( X ) , ≤ s ) (ev en for an a rbitrary T ych onoff space ( X , τ )) is g iv en only in [3, Theorem 5.9] where this is done b y means of t he in tro duced there sp ecial kind of pro ximities, called LC-proximities . Definition 3.22 Let ( X , τ ) b e a T yc ho noff space. An NCA ( RC ( X , τ ) , C ) is said to b e admis s ible for ( X, τ ) if it satisfies the following conditions: (AK1) if F , G ∈ RC ( X ) and F ∩ G 6 = ∅ then F C G (i.e. ρ X ⊆ C ); (AK2) if F ∈ RC ( X ) and x ∈ int X ( F ) then there exists G ∈ RC ( X ) suc h that x ∈ int X ( G ) and G ≪ C F . The set of all NCAs whic h a re a dmissible for ( X , τ ) will b e denoted b y K ad ( X , τ ) (or simply by K ad ( X )). If ( R C ( X ) , C i ) ∈ K ad ( X ), where i = 1 , 2, then we set ( RC ( X ) , C 1 )  ad ( RC ( X ) , C 2 ) iff C 2 ⊆ C 1 . Corollary 3.23 ([2]) I f X is a T ychonoff sp ac e then the o r de r e d se ts ( K ( X ) , ≤ ) and ( K ad ( X ) ,  ad ) ar e isomorphic. Pr o o f. It follows immediately from Theorem 3.20. Recall that O. F rink [8] intro duced the not io n of a W a llma n- t ype compactifi- cation a nd aske d whether eve ry Hausdorff compactification of a T yc honoff space is a W allman-type compactification. This que stion was a nswe red in negativ e b y V. M. Ul’jano v [18 ]. W e will giv e a necessary and sufficien t condition f o r a compactification of a discrete space to b e of a W allman t yp e (recall that, according to the Reduction Theorem of L. B. ˇ Sapiro [13 ] (see also [16]), it is enough to in ves tigate the F rink’s problem o nly in the class of discrete spaces). Our criterion follows easily from the follo wing result of O. Nj ˚ astad: Theorem 3.24 ([12]) L et ( X , τ ) b e a T ychon o ff sp ac e and ( cX , c ) b e a c o m p act- ific ation of X . L et δ c b e the Efr emoviˇ c pr oximity on the sp ac e ( X , τ ) de fi ne d as fol lows: for a ny M , N ⊆ X , M δ c N iff cl cX ( c ( M )) ∩ cl cX ( c ( N )) 6 = ∅ . Then ( cX , c ) is a Wal lman -typ e c omp actific ation of ( X , τ ) i f and only if ther e exists a family B 18 of close d subsets of ( X , τ ) whi c h is close d under finite interse ction s and satisfies the fol lowing two c o nditions: (B1) If F , G ∈ B and F ∩ G = ∅ then F ( − δ c ) G ; (B2) I f A, B ⊆ X and A ( − δ c ) B then ther e exist F , G ∈ B such that A ⊆ F , B ⊆ G and F ∩ G = ∅ . Prop osition 3.25 L et X b e a discr ete sp ac e, Ψ t ( X ) = ( A, ρ, I B) and C ∈ K a ( X ) . Then µ − 1 c ( C ) (se e Cor ol lary 3.11 for µ c ) is a Wal lman-typ e c omp a c tific ation of X iff ther e exists a sub-me et-semilattic e B o f A such that: (1) for any a, b ∈ B , aρb iff aC b , and (2) for any a, c ∈ A , a ≪ C c implies that ther e ex ists b ∈ B such that a ≤ b ≪ C c . Pr o o f. Note that condition (2) ma y b e substituted for the following one: (2 ′ ) for an y a, c ∈ A , a ≪ C c implies that there exist b 1 , b 2 ∈ B such that a ≤ b 1 ≤ b ∗ 2 ≤ c . No w, applying Remark 3 .9 a nd Theorem 3.24 , w e complete the pro of. References [1] W. Comfort, S. Negrep on tis, C hain Cond itions in T op ology , Cam bridge Univ. Press, Cam b ridge, 1982. [2] H. d e V ries, Compact Spaces and Comp actificatio ns, an Algebraic Appr oac h, V an Gorcum, T he Netherland s, 19 62. [3] G. Dimo v, Regular and other kinds of extensions of top ological sp aces, Serdica Math. J. 24 (1998 ) 99–126 . [4] G. Dimo v, Some generalizations of F edorch uk Dualit y Theorem - I, arXiv:0709. 4495. [5] G. Dimo v, D. V ak arelo v, Contact algebras and region-based theory of space: a pro ximit y approac h – I, F und amen ta Informaticae 74(2-3) (2006) 20 9–249 . [6] R. Engelking, General T op ology , PWN, W ars zaw a, 1977. [7] V. V. F edorc huk, Bo olean δ -algebras and q u asi-op en mappings, S ibirsk. Mat. ˇ Z. 14 (5) (1973) 1088–1 099 = Siber ian Math. J . 14 (197 3), 759–7 67 (1974). [8] O. F rin k , Compactificatio ns and semi-normal spaces, A mer. J. Math. 86 (1964) 602–6 07. [9] P . T. John stone, S tone S p aces, Cam bridge Un iv. Press, Cambridge, 1982. [10] S. L eader, Lo cal pro ximit y spaces, Math. Annalen 1 69 (19 67) 275– 281. [11] J. Miod uszewski, L. Ru dolf, H-closed and extremally disconected Hausdorff spaces, Dissert. M ath. (Ro zpr. Mat.) 66 (196 9) 1–52. 19 [12] O. Nj ˚ astad, On W allman-t y p e compactifications, Math. Zeitsc hr. 91 (1966) 267–2 76. [13] L. B. ˇ Sapiro, A redu ction of the b asic p r oblem on compactifications of W allman t yp e, Dokl. Ak ad. Nauk SSSR 217 (1974 ) 38–41 = So viet Math. Dokl. 15 (1974) 1020–1 023. [14] R. S ikorski, Boolean Algebras, Spr inger-V erlag, Be rlin, 1964 . [15] J. M. Smirnov, On p ro xim ity spaces, Mat. Sb. 31 (19 52) 543– 574. [16] A. K. Steiner, E. F. S teiner, On the reduction of the W allman compactification problem to discrete spaces, General T op ology and Appls. 7 (1977) 35–37. [17] P . V op ˇ enk a, On the d im en sion of compact spaces, C zechosl. Math. J. 8 (1958) 319–3 27. [18] V. M. Ul’jano v, Solution of a basic p roblem on compactificatio ns of W allman t yp e, So viet Math. D okl. 18 (1977) 567 –571. [19] D. V ak arelo v, G. Dimov, I . D ¨ un tsc h, B. Bennett, A pr o ximity app roac h to some region-based theories of sp ace, J. App lied Non-Classical Logic s 12 (3-4) (20 02) 52 7– 559. 20