De Vries duality for normal spaces and locally compact Hausdorff spaces
aa r X i v : . [ m a t h . GN ] A p r DE VRIES DUALITY FOR NORMAL SPACES AND LOCALLYCOMPACT HAUSDORFF SPACES
G. BEZHANISHVILI, P. J. MORANDI, B. OLBERDING
Abstract.
By de Vries duality, the category of compact Hausdorff spaces is dually equiv-alent to the category of de Vries algebras. In our recent article, we have extended de Vriesduality to completely regular spaces by generalizing de Vries algebras to de Vries extensions.To illustrate the utility of this point of view, we show how to use this new duality to obtainalgebraic counterparts of normal and locally compact Hausdorff spaces in the form of deVries extensions that are subject to additional axioms which encode the desired topologicalproperty. This, in particular, yields a different perspective on de Vries duality. As a furtherapplication, we show that a duality for locally compact Hausdorff spaces due to Dimov canbe obtained from our approach. Introduction
It is a well-known theorem of Smirnov that compactifications of a completely regularspace X can be described “internally” by means of proximities on X compatible with thetopology on X (see, e.g., [9, Sec. 7]), where a proximity is a binary relation on the powersetof X satisfying certain natural axioms, including a point-separation axiom (see, e.g., [9,Sec. 3]). De Vries takes this further in [2] by axiomatizing the proximities on the completeBoolean algebra of regular open subsets of X that correspond to compactifications of X .In de Vries’ work, the point-separation axiom is replaced by the point-free axiom assertingthat the proximity relation is approximating. This point-free axiom decouples the proximityfrom the underlying space and yields what is known today as a de Vries algebra: a completeBoolean algebra with a binary relation satisfying all of the axioms of a proximity except thepoint-separation axiom, which is replaced by de Vries’ point-free axiom. De Vries showedthat this axiomatization can be used to give an algebraic description of the category KHaus of compact Hausdorff spaces. More formally, the category
KHaus is dually equivalent to thecategory
DeV of de Vries algebras.In [1] we extended de Vries duality to completely regular spaces by generalizing the notionof a de Vries algebra to that of a de Vries extension. While de Vries duality alone is notenough to deal with completely regular spaces and de Vries extensions, we show in [1] that deVries duality together with Tarski duality for complete and atomic Boolean algebras providesan appropriate framework for dealing with completely regular spaces. In fact, the methods
Mathematics Subject Classification.
Key words and phrases.
Compact Hausdorff space; normal space; locally compact space; proximity; deVries duality. in [1] yield a dual equivalence between the category of de Vries extensions and the categoryof compactifications of completely regular spaces that extends both de Vries duality andTarski duality. The de Vries extensions corresponding to Stone- ˇCech compactifications areaxiomatized in [1] as “maximal” de Vries extensions. This in turn yields a dual equivalencebetween the category of completely regular spaces and the category of maximal de Vriesextensions, thereby providing an algebraic counterpart to completely regular spaces.It is noted in [1] that discrete spaces can be viewed algebraically as trivial de Vries ex-tensions. The interpretation of more interesting classes of completely regular spaces is notas straightforward. In this article, we continue our work begun in [1] by giving algebraicinterpretations of normal spaces and locally compact Hausdorff spaces within our frameworkof de Vries extensions. On a technical level, this involves, for a compactification e ∶ X → Y of a completely regular space X , a close analysis of the corresponding de Vries extension e − ∶ (RO( Y ) , ≺) → (℘( X ) , ⊆) , where RO( Y ) is the complete Boolean algebra of regularopen sets of Y and ℘( X ) is the powerset of X . Some of the main results of the current paperinvolve determining which algebraic properties of the map e − reflect the normality and localcompactness of X . With these characterizations, we obtain dual equivalences between thecategories of such spaces and the appropriate full subcategories of the category of de Vriesextensions.The article is organized as follows. In Section 2 we recall all the necessary backgroundabout de Vries algebras, de Vries extensions, and maximal de Vries extensions. In Section 3we introduce normal de Vries extensions and show that every normal de Vries extension ismaximal. This allows us to view normal de Vries extensions as a full subcategory NDeVe of the category
MDeVe of maximal de Vries extensions. We prove that
NDeVe is duallyequivalent to the category
Norm of normal spaces. In Section 4 we introduce locally compactde Vries extensions. While not every locally compact de Vries extension is maximal, weprove that
LKHaus is dually equivalent to the full subcategory
LDeVe of MDeVe consistingof locally compact de Vries extensions.In Section 5 we introduce compact de Vries extensions. Since every compact de Vriesextension is maximal, we view compact de Vries extensions as a full subcategory
CDeVe of MDeVe . We prove that
CDeVe is equivalent to
DeV , and hence dually equivalent to
KHaus .This gives another perspective on de Vries duality. In Section 6 we introduce minimal deVries extensions and show that non-compact minimal de Vries extensions correspond toone-point compactifications of non-compact locally compact Hausdorff spaces.While we do not make use of it directly, we are motivated in our treatment of local com-pactness by Leader’s generalization in [7] of a proximity relation to that of a local proximityrelation, a generalization that yields a description of the local compactifications of a com-pletely regular space by means of local proximity relations compatible with the topology.Recently, Dimov [3] has recast Leader’s work in a setting similar to de Vries algebras, andobtained a duality theorem for the category
LKHaus of locally compact Hausdorff spaces
E VRIES DUALITY FOR NORMAL SPACES AND LOCALLY COMPACT HAUSDORFF SPACES 3 that generalizes de Vries duality. In Section 7 we show that Dimov’s duality for
LKHaus canbe derived as a consequence of our duality for
LKHaus .2.
De Vries extensions
In this preliminary section we recall de Vries algebras, de Vries extensions, and maximal deVries extensions. By de Vries duality [2], de Vries algebras provide an algebraic counterpartof compact Hausdorff spaces. By the duality developed in [1], de Vries extensions provide analgebraic counterpart of compactifications of completely regular spaces. Under this duality,maximal de Vries extensions correspond to Stone- ˇCech compactifications, thus yielding aduality for completely regular spaces that generalizes de Vries duality for compact Hausdorffspaces.2.1.
De Vries algebras and compact Hausdorff spaces.
We start by recalling de Vriesalgebras and de Vries morphisms.
Definition 2.1. (1) A de Vries algebra is a pair A = ( A, ≺ ) , where A is a complete Boolean algebra and ≺ is a binary relation on A satisfying the following axioms:(DV1) 1 ≺ a ≺ b implies a ≤ b .(DV3) a ≤ b ≺ c ≤ d implies a ≺ d .(DV4) a ≺ b, c implies a ≺ b ∧ c .(DV5) a ≺ b implies ¬ b ≺ ¬ a .(DV6) a ≺ b implies there is c such that a ≺ c ≺ b .(DV7) b = ⋁{ a ∈ A ∣ a ≺ b } .(2) A de Vries morphism is a map ρ ∶ A → B between de Vries algebras satisfying thefollowing axioms:(M1) ρ ( ) = ρ ( a ∧ b ) = ρ ( a ) ∧ ρ ( b ) .(M3) a ≺ b implies ¬ ρ (¬ a ) ≺ ρ ( b ) .(M4) ρ ( b ) = ⋁{ ρ ( a ) ∣ a ≺ b } .De Vries algebras and de Vries morphisms form a category DeV where the composition oftwo de Vries morphisms ρ ∶ A → A and ρ ∶ A → A is defined by ( ρ ⋆ ρ )( b ) = ⋁{ ρ ρ ( a ) ∣ a ≺ b } . De Vries algebras arise naturally from compact Hausdorff spaces. If X is a compactHausdorff space, then the pair X ∗ = (RO( X ) , ≺ ) is a de Vries algebra, where RO( X ) isthe complete Boolean algebra of regular open subsets of X and ≺ is the canonical proximityrelation on RO( X ) given by U ≺ V iff cl ( U ) ⊆ V. G. BEZHANISHVILI, P. J. MORANDI, B. OLBERDING
Similarly, de Vries morphisms arise naturally from continuous maps between compact Haus-dorff spaces. If f ∶ X → Y is such a map, then f ∗ ∶ RO( Y ) → RO( X ) is a de Vries morphism,where f ∗ is given by f ∗ ( U ) = int ( cl ( f − ( U ))) . This defines a contravariant functor (−) ∗ ∶ KHaus → DeV . To define a contravariant functor (−) ∗ ∶ DeV → KHaus , let A be a de Vries algebra. For S ⊆ A , let ↟ S = { a ∈ A ∣ b ≺ a for some b ∈ S } and ↡ S = { a ∈ A ∣ a ≺ b for some b ∈ S } . A filter F of A is round if ↟ F = F . Similarly, an ideal I of A is round if ↡ I = I .An end of A is a maximal proper round filter. Let Y A be the set of ends of A . For a ∈ A ,set ζ A ( a ) = { x ∈ Y A ∣ a ∈ x } , and define a topology on Y A by letting ζ A [ A ] = { ζ A ( a ) ∣ a ∈ A } be a basis for the topology. Then Y A is compact Hausdorff, and we set A ∗ = Y A . For a deVries morphism ρ ∶ A → A ′ , define ρ ∗ ∶ Y A ′ → Y A by ρ ∗ ( y ) = ↟ ρ − ( y ) . Then ρ ∗ is a well-definedcontinuous map, yielding a contravariant functor ( − ) ∗ ∶ DeV → KHaus .We have that ζ A ∶ A → ( A ∗ ) ∗ is a de Vries isomorphism, and ξ X ∶ X → ( X ∗ ) ∗ is ahomeomorphism, where ξ ( x ) = { U ∈ X ∗ ∣ x ∈ U } . Therefore, ζ ∶ DeV → ( − ) ∗ ○ ( − ) ∗ and ξ ∶ KHaus → ( − ) ∗ ○ ( − ) ∗ are natural isomorphisms. Thus, we arrive at de Vries duality. Theorem 2.2. (de Vries [2])
KHaus is dually equivalent to
DeV . De Vries extensions and compactifications.
We next generalize de Vries algebrasto de Vries extensions [1]. For this we will utilize Tarski duality between the category
CABA of complete and atomic Boolean algebras with complete Boolean homomorphismsand the category
Set of sets and functions. If X is a set, then ℘ ( X ) is a complete andatomic Boolean algebra, and if f ∶ X → Y is a function, then f − ∶ ℘ ( Y ) → ℘ ( X ) is acomplete Boolean homomorphism. This yields a contravariant functor Set → CABA . Goingbackwards, for a complete and atomic Boolean algebra B , let X B be the set of atoms of B ,and for a complete Boolean homomorphism σ ∶ B → B , let σ + ∶ X B → X B be given by σ + ( x ) = ⋀ { b ∈ B ∣ x ≤ σ ( b )} . It is well known that σ + is a well-defined function, yieldinga contravariant functor CABA → Set . For each set X , we have a natural isomorphism η X ∶ X → X ℘( X ) , given by η X ( x ) = { x } for each x ∈ X ; and for each B ∈ CABA , we have anatural isomorphism ϑ B ∶ B → ℘ ( X B ) , given by ϑ B ( b ) = { x ∈ X B ∣ x ≤ b } . Definition 2.3. (1) A de Vries algebra A = ( A, ≺ ) is extremally disconnected if a ≺ b iff a ≤ b . E VRIES DUALITY FOR NORMAL SPACES AND LOCALLY COMPACT HAUSDORFF SPACES 5 (2) A de Vries algebra A = ( A, ≺ ) is atomic if A is atomic as a Boolean algebra.(3) A de Vries extension is a 1-1 de Vries morphism α ∶ A → B such that A is a deVries algebra, B is an atomic extremally disconnected de Vries algebra, and α [ A ] isjoin-meet dense in B .A morphism between de Vries extensions α ∶ A → B and α ′ ∶ A ′ → B ′ is a pair ( ρ, σ ) ,where ρ ∶ A → A ′ is a de Vries morphism, σ ∶ B → B ′ is a complete Boolean homomorphism,and σ ○ α = α ′ ⋆ ρ . A α / / ρ (cid:15) (cid:15) B σ (cid:15) (cid:15) A ′ α ′ / / B ′ Since σ is a complete Boolean homomorphism and B , B ′ are extremally disconnected, σ is a de Vries morphism and σ ⋆ α = σ ○ α ; hence, if ( ρ, σ ) is a morphism in DeVe , then thediagram above commutes in
DeV (see [1, Rems. 2.6, 4.10]).The composition of two morphisms ( ρ , σ ) and ( ρ , σ ) is defined as ( ρ ⋆ ρ , σ ○ σ ) . A ρ ⋆ ρ (cid:26) (cid:26) α / / ρ (cid:15) (cid:15) B σ (cid:15) (cid:15) σ ○ σ (cid:4) (cid:4) A α / / ρ (cid:15) (cid:15) B σ (cid:15) (cid:15) A α / / B It is straightforward to see that de Vries extensions with morphisms between them form acategory, which we denote
DeVe .De Vries extensions arise naturally from compactifications of completely regular spaces.Let e ∶ X → Y be a compactification of a completely regular space X . Then ( RO ( Y ) , ≺ ) is a de Vries algebra, the powerset ( ℘ ( X ) , ⊆ ) is an atomic extremally disconnected de Vriesalgebra, and the pullback map e − ∶ RO ( Y ) → ℘ ( X ) is a de Vries extension.Let Comp be the category whose objects are compactifications e ∶ X → Y and whosemorphisms are pairs ( f, g ) of continuous maps such that the following diagram commutes. X e / / f (cid:15) (cid:15) Y g (cid:15) (cid:15) X ′ e ′ / / Y ′ That is, each element of B is a join of meets of elements of α [ A ] . This is equivalent to each element of B being a meet of joins of elements of α [ A ] (see [1, Rem. 4.7]). G. BEZHANISHVILI, P. J. MORANDI, B. OLBERDING
The composition of two morphisms ( f , g ) and ( f , g ) in Comp is ( f ○ f , g ○ g ) . X e / / f ○ f (cid:26) (cid:26) f (cid:15) (cid:15) Y g (cid:15) (cid:15) g ○ g (cid:4) (cid:4) X e / / f (cid:15) (cid:15) Y g (cid:15) (cid:15) X e / / Y For a morphism ( f, g ) in Comp , the pair ( g ∗ , f − ) is a morphism in DeVe , where g ∗ is thede Vries dual of g . RO ( Y ′ ) g ∗ (cid:15) (cid:15) ( e ′ ) − / / ℘ ( X ′ ) f − (cid:15) (cid:15) RO ( Y ) e − / / ℘ ( X ) This yields a contravariant functor E ∶ Comp → DeVe . To define a contravariant functor C ∶ DeVe → Comp , let α ∶ A → B be a de Vries extension. Let X B be the set of atoms of B .For b ∈ X B , we have ↑ b is an ultrafilter of B , and we define α ∗ ∶ X B → Y A by α ∗ ( b ) = ↟ α − ( ↑ b ) . We can view X B as a subset of Y B by sending b to ↑ b . Then we can think of α ∗ as therestriction to X B of the de Vries dual α ∗ ∶ Y B → Y A . By [1, Lem. 5.4], α ∗ is 1-1. Let τ α be the least topology on X B making α ∗ continuous. By [1, Thm. 5.7], α ∗ ∶ X B → A is acompactification. For a morphism ( ρ, σ ) in DeVe A α / / ρ (cid:15) (cid:15) B σ (cid:15) (cid:15) A ′ α ′ / / B ′ the pair ( σ + , ρ ∗ ) is a morphism in Comp , X B ′ α ′∗ / / σ + (cid:15) (cid:15) Y A ′ ρ ∗ (cid:15) (cid:15) X B α ∗ / / Y A where ρ ∗ is the de Vries dual of ρ and σ + is the Tarski dual of σ . This yields a contravariantfunctor C ∶ DeVe → Comp , and the functors E and C establish a dual equivalence between Comp and
DeVe : Theorem 2.4. [1, Thm. 5.9]
Comp is dually equivalent to
DeVe . E VRIES DUALITY FOR NORMAL SPACES AND LOCALLY COMPACT HAUSDORFF SPACES 7
Maximal de Vries extensions, Stone- ˇCech compactifications, and completelyregular spaces.
We next turn to maximal de Vries extensions.
Definition 2.5. (1) We call two de Vries extensions α ∶ A → B and γ ∶ C → B compatible if α [ A ] = γ [ C ] .(2) We say that a de Vries extension α ∶ A → B is maximal provided for every compatiblede Vries extension γ ∶ C → B , there is a de Vries morphism δ ∶ C → A such that α ⋆ δ = γ . A α / / BC δ _ _ ❄❄❄❄❄❄❄❄ γ ? ? ⑦⑦⑦⑦⑦⑦⑦⑦ (3) Let MDeVe be the full subcategory of
DeVe consisting of maximal de Vries extensions.
Remark 2.6.
Let α ∶ A → B and γ ∶ C → B be compatible de Vries extensions. Since α and γ are 1-1 with the same image, we have a bijection δ ∶ = α − ○ γ ∶ C → A with inverse γ − ○ α ∶ A → C . Because α and γ are both 1-1 and meet preserving, they are both orderpreserving and order reflecting, so δ and its inverse are poset isomorphisms, hence Booleanisomorphisms. Theorem 2.7. [1, Thm. 6.4] If e ∶ X → Y is a compactification, then the associated de Vriesextension e − ∶ RO ( Y ) → ℘ ( X ) is maximal iff e ∶ X → Y is isomorphic in Comp to theStone- ˇCech compactification s ∶ X → βX . Remark 2.8.
Equivalent compactifications e ∶ X → Y and e ′ ∶ X → Y ′ are isomorphic in Comp but the converse is not true in general [1, Ex. 3.2]. However, if e ′ is the Stone- ˇCechcompactification of X , then e is equivalent to e ′ iff e is isomorphic to e ′ in Comp [1, Thm. 3.3].Let
CReg be the category of completely regular spaces and continuous maps. Sendinga completely regular space X to its Stone- ˇCech compactification s ∶ X → βX yields anequivalence between CReg and the full subcategory of
Comp consisting of Stone- ˇCech com-pactifications. Since Stone- ˇCech compactifications dually correspond to maximal de Vriesextensions, we arrive at the following duality theorem, which generalizes de Vries duality tocompletely regular spaces.
Theorem 2.9. [1, Thm. 6.9]
CReg is dually equivalent to
MDeVe . Additional properties of de Vries algebras and de Vries extensions.
We con-clude this preliminary section by recalling some basic facts about de Vries algebras and deVries extensions that we will use subsequently. We start with de Vries algebras.
Lemma 2.10.
Let A be a de Vries algebra and Y A its dual compact Hausdorff space. (1) There is an isomorphism between the lattice of round filters of A (ordered by reverseinclusion) and the lattice of round ideals (ordered by inclusion), given by F ↦ { a ∈ A ∣ ¬ a ∈ F } for a round filter F and I ↦ { a ∈ A ∣ ¬ a ∈ I } for a round ideal I . G. BEZHANISHVILI, P. J. MORANDI, B. OLBERDING (2)
There is an isomorphism between the lattice of round filters of A and the lattice ofclosed subsets of Y A , given by F ↦ ⋂ { ζ A ( a ) ∣ a ∈ F } and C ↦ { a ∈ A ∣ C ⊆ ζ A ( a )} . (3) There is an isomorphism between the lattice of round ideals of A and the lattice ofopen subsets of Y A , given by I ↦ ⋃ { ζ A ( a ) ∣ a ∈ I } and U ↦ { a ∈ A ∣ cl ( ζ A ( a )) ⊆ U } . The proof of Lemma 2.10(1) is straightforward, that of Lemma 2.10(2) is given in [2,Thm. 1.3.12], and Lemma 2.10(3) is proved similarly. The proof of the next lemma is alsostraightforward, and we skip it.
Lemma 2.11.
Let ρ ∶ A → A ′ be a de Vries morphism. (1) If a ∈ A , then ρ ( a ) ≤ ¬ ρ ( ¬ a ) . (2) If I is an ideal of A , a , . . . , a n ∈ I , x ∈ A ′ , and x ≤ ρ ( a ) ∨ ⋯ ∨ ρ ( a n ) , then there is b ∈ I ( b = a ∨ ⋯ ∨ a n ) such that x ≤ ρ ( b ) . The next two lemmas are about de Vries extensions.
Lemma 2.12. [1, Lem. 4.1]
Let e ∶ X → Y be a compactification of a completely regularspace X . If e − ∶ RO ( Y ) → ℘ ( X ) is the corresponding de Vries extension, then the image of e − is RO ( X ) and e − is a Boolean isomorphism from RO ( Y ) to RO ( X ) . Lemma 2.13.
Let α ∶ A → B be a de Vries extension. (1) [1, Lem. 5.3] If a ∈ A and b ∈ X B , then b ≤ α ( a ) iff α ∗ ( b ) ∈ ζ ( a ) . (2) [1, Lem. 6.2] If γ ∶ C → B is another de Vries extension, then α and γ are compatibleiff they induce the same topology on X B . Normal de Vries extensions
In this section we introduce normal de Vries extensions and show that every normal deVries extension is maximal. We prove that the dual equivalence of Theorem 2.9 between
CReg and
MDeVe restricts to a dual equivalence between the full subcategory
Norm of CReg consisting of normal spaces and the full subcategory
NDeVe of MDeVe consisting of normalde Vries extensions.
Definition 3.1. (1) We call a de Vries extension α ∶ A → B normal provided the following axiom holds:If F is a round filter and I a round ideal of A with ⋀ α [ F ] ≤ ⋁ α [ I ] , then there are a, b ∈ A such that a ≺ b , ⋀ α [ F ] ≤ α ( a ) , and α ( b ) ≤ ⋁ α [ I ] .(2) Let NDeVe be the full subcategory of
DeVe consisting of normal de Vries extensions.It is a well-known theorem (see, e.g., [4, Cor. 3.6.4]) that a compactification e ∶ X → Y of anormal space is equivalent to the Stone- ˇCech compactification s ∶ X → βX iff disjoint closed E VRIES DUALITY FOR NORMAL SPACES AND LOCALLY COMPACT HAUSDORFF SPACES 9 sets in X have disjoint closures in Y . We next prove an algebraic version of this result bycharacterizing normal de Vries extensions. Theorem 3.2.
Let e ∶ X → Y be a compactification. Then the associated de Vries extension e − ∶ RO ( Y ) → ℘ ( X ) is normal iff X is normal and e is isomorphic to the Stone- ˇCechcompactification s ∶ X → βX of X . In particular, X is normal iff s − ∶ RO ( βX ) → ℘ ( X ) isa normal de Vries extension.Proof. First suppose that X is normal and e is isomorphic to the Stone- ˇCech compactification s ∶ X → βX . To simplify notation, we identify e with s and view X as a dense subspace of βX . Then s − ( W ) = W ∩ X . Let F be a round filter and I a round ideal of RO ( βX ) with ⋀ s − [ F ] ≤ ⋁ s − [ I ] . We have ⋀ s − [ F ] = ( ⋂ F ) ∩ X = ∶ C is closed in X , ⋁ s − [ I ] = ( ⋃ I ) ∩ X = ∶ U is open in X , and C ⊆ U . Let D = X ∖ U . Then D is closed and C ∩ D = ∅ . Using that X is normal twice, there are V , V ∈ RO ( X ) with C ⊆ V , D ⊆ V , and cl X ( V ) ∩ cl X ( V ) = ∅ . By Lemma 2.12, there are W , W ∈ RO ( βX ) with W i ∩ X = V i . By [4, Cor. 3.6.4], cl βX ( V ) ∩ cl βX ( V ) = ∅ , so since X is dense in βX , we have cl βX ( W ) ∩ cl βX ( W ) = ∅ .Therefore, cl βX ( W ) ⊆ βX ∖ cl βX ( W ) . Since W is regular open, cl βX ( W ) is regular closed,so W ∶ = βX ∖ cl βW ( W ) is regular open and W ≺ W . We have C ⊆ V = W ∩ X and W ∩ X = X ∩ ( βX ∖ cl βX ( W )) = X ∖ cl βX ( W ) = X ∖ cl βX ( W ∩ X ) = X ∖ cl X ( V ) ⊆ X ∖ V ⊆ U. Therefore, we have found W ≺ W with ⋀ s − [ F ] ⊆ W and W ⊆ ⋁ s − [ I ] . This proves that s − ∶ RO ( βX ) → ℘ ( X ) is a normal de Vries extension.Conversely, suppose that e − ∶ RO ( Y ) → ℘ ( X ) is a normal de Vries extension. By [4,Cor. 3.6.4], to show that X is normal and e is isomorphic to the Stone- ˇCech compactificationof X it is sufficient to show that if C and D are disjoint closed sets of X , then cl Y ( C ) ∩ cl Y ( D ) = ∅ . Let U = X ∖ D . Then C ⊆ U . Set V = int Y ( U ∪ ( Y ∖ X )) . By Lemma 2.10,there is a round filter F of RO ( Y ) with ⋂ F = cl Y ( C ) and a round ideal I of RO ( Y ) with ⋃ I = V . Therefore, ⋀ e − [ F ] = cl Y ( C ) ∩ X = C and ⋁ e − [ I ] = V ∩ X = U . Since e − ∶ RO ( Y ) → ℘ ( X ) is a normal de Vries extension, there are W , W ∈ RO ( Y ) with W ≺ W , C ⊆ W ∩ X , and W ∩ X ⊆ U . Then cl Y ( W ) ⊆ W . Since C ⊆ W ∩ X ⊆ W , we seethat cl Y ( C ) ⊆ cl Y ( W ) ⊆ W . Also, W ∩ X ⊆ U , so W ⊆ U ∪ ( Y ∖ X ) . Since W is open in Y ,it follows that W ⊆ int Y ( U ∪ ( Y ∖ X )) = V . Therefore, cl Y ( C ) ⊆ V , so cl Y ( C ) ∩ ( Y ∖ V ) = ∅ .Finally, Y ∖ V = Y ∖ int Y ( U ∪ ( Y ∖ X )) = cl Y ( X ∖ U ) = cl Y ( D ) . Consequently, cl Y ( C ) ∩ cl Y ( D ) = ∅ . This finishes the proof. (cid:3) Corollary 3.3.
Let α ∶ A → B be a normal de Vries extension. Then α is maximal.Proof. This follows from Theorems 2.7 and 3.2 (cid:3)
Remark 3.4.
Using Theorem 2.4, we can phrase Theorem 3.2 dually as follows: A de Vriesextension α ∶ A → B is normal iff X B is normal and the corresponding compactification α ∗ ∶ X B → Y A is isomorphic to the Stone- ˇCech compactification of X B .Let Norm be the full subcategory of
CReg consisting of normal spaces. Putting Theo-rems 2.9 and 3.2 together yields the following duality theorem for normal spaces.
Theorem 3.5.
There is a dual equivalence between
Norm and
NDeVe . Locally compact de Vries extensions
In this section we introduce locally compact de Vries extensions. Unlike normal de Vriesextensions, locally compact de Vries extensions do not have to be maximal. We prove thatthe category
LDeVe of locally compact maximal de Vries extensions is dually equivalent tothe category
LKHaus of locally compact Hausdorff spaces.Let e ∶ X → Y be a compactification and e − ∶ RO ( Y ) → ℘ ( X ) the corresponding de Vriesextension. For U ∈ RO ( Y ) , we have: ¬ e − ( ¬ U ) = X ∖ e − ( int Y ( Y ∖ U )) = X ∖ e − ( Y ∖ cl Y ( U )) = X ∖ ( X ∖ e − ( cl Y ( U ))) = e − ( cl Y ( U )) . Therefore, ¬ e − ( ¬ U ) is a closed subset of X . For it to be compact, since e − [ RO ( Y )] = RO ( X ) by Lemma 2.12, if ¬ e − ( ¬ U ) ⊆ ⋃ { e − ( V i ) ∣ i ∈ I } , with V i ∈ RO ( Y ) , then there is afinite J ⊆ I with ¬ e − ( ¬ U ) ⊆ ⋃ { e − ( V i ) ∣ i ∈ J } . This motivates the following definition. Definition 4.1.
Let α ∶ A → B be a de Vries extension.(1) We call a ∈ A α -compact provided ¬ α ( ¬ a ) ≤ ⋁ α [ S ] for some S ⊆ A implies that thereis a finite T ⊆ S with ¬ α ( ¬ a ) ≤ ⋁ α [ T ] .(2) Let I α = { a ∈ A ∣ a is α -compact } . Lemma 4.2. If α ∶ A → B is a de Vries extension, then I α is an ideal of A .Proof. First, 0 ∈ I α because ¬ α ( ¬ ) = ¬ α ( ) = ¬ = α -compact, so I α is nonempty.Next, let a ∈ I α and b ≤ a . Suppose that ¬ α ( ¬ b ) ≤ ⋁ α [ S ] for some S ⊆ A . Then ¬ α ( ¬ a ) ≤ = ¬ α ( ¬ b ) ∨ α ( ¬ b ) ≤ ⋁ α [ S ] ∨ α ( ¬ b ) , so there is a finite T ⊆ S with ¬ α ( ¬ a ) ≤ ⋁ α [ T ] ∨ α ( ¬ b ) . Since b ≤ a , we have ¬ α ( ¬ b ) ≤ ¬ α ( ¬ a ) ,so ¬ α ( ¬ b ) ≤ ⋁ α [ T ] , which shows that b ∈ I α . Finally, let a, b ∈ I α . Suppose ¬ α ( ¬ ( a ∨ b )) ≤ ⋁ α [ S ] . Then ¬ α ( ¬ a ) ∨ ¬ α ( ¬ b ) = ¬ α ( ¬ ( a ∨ b )) ≤ ⋁ α [ S ] . Therefore, there are finite T, T ′ ⊆ S with ¬ α ( ¬ a ) ≤ ⋁ α [ T ] and ¬ α ( ¬ b ) ≤ ⋁ α [ T ′ ] . Thus, ¬ α ( ¬ ( a ∨ b )) ≤ ⋁ α [ T ∪ T ′ ] , yieldingthat a ∨ b ∈ I α . This completes the proof that I α is an ideal of A . (cid:3) Theorem 4.3.
Let α ∶ A → B be a de Vries extension. Then the following are equivalent. (1) X B is locally compact. (2) For each b ∈ A we have α ( b ) = ⋁ { α ( a ) ∣ a ∈ I α , a ≺ b } . E VRIES DUALITY FOR NORMAL SPACES AND LOCALLY COMPACT HAUSDORFF SPACES 11 (3) I α is a round ideal with ⋁ α [ I α ] = .Proof. (1) ⇒ (2). Suppose that X B is locally compact. The de Vries extension α ∶ A → B isisomorphic to α − ∗ ∶ RO ( Y A ) → ℘ ( X B ) . Under this isomorphism, I α corresponds to I ∶ = I α − ∗ .To simplify notation we view X B as a dense subspace of Y A . Let V ∈ RO ( Y A ) . Then themap α − ∗ sends V to V ∩ X B , so V ∩ X B is the union of those U ∈ RO ( X B ) for which cl X B ( U ) ⊆ V ∩ X B and is compact. Since cl X B ( U ) is compact, cl X B ( U ) is closed in Y A . Thus, cl Y A ( U ) = cl X B ( U ) . By Lemma 2.12, U = W ∩ X B for some W ∈ RO ( Y A ) . Because X B isdense in Y A , we have cl Y A ( W ) = cl Y A ( U ) , which yields cl Y A ( W ) ⊆ V ∩ X B ⊆ V . Therefore, W ≺ V . Moreover, as cl Y A ( W ) ∩ X B = cl Y A ( W ) = cl X B ( U ) is compact, W ∈ I . Thus, V ∩ X B is the union of W ∩ X B for W ∈ I and W ≺ V , which yields (2).(2) ⇒ (3). By Lemma 4.2, I α is an ideal. By (2), α ( ) = ⋁ { α ( a ) ∣ a ∈ I α } , so 1 = ⋁ α [ I α ] .To see that I α is round, let a ∈ I α . Then ¬ α ( ¬ a ) ≤ = ⋁ α [ I α ] . Since a is α -compact, thereare a , . . . , a n ∈ I α with ¬ α ( ¬ a ) ≤ α ( a ) ∨ ⋯ ∨ α ( a n ) . By Lemma 2.11, there is b ∈ I α with ¬ α ( ¬ a ) ≤ α ( b ) . By (2), α ( b ) = ⋁ { α ( c ) ∣ c ∈ I α , c ≺ b } , so repeating the above argument with1 replaced by b yields c ≺ b with ¬ α ( ¬ a ) ≤ α ( c ) . Thus, α ( a ) ≤ ¬ α ( ¬ a ) ≤ α ( c ) , so a ≤ c , andhence a ≺ b . Since b ∈ I α , this shows that I α is round.(3) ⇒ (1). Since I α is a round ideal, by Lemma 2.10, it corresponds to the open subset U ∶ = ⋃ { ζ ( a ) ∣ a ∈ I α } of Y A . Claim 4.4. If U = ⋃ { ζ ( a ) ∣ a ∈ I α } , then U = α ∗ [ X B ] .Proof of the Claim . Let b be an atom of B . Since ⋁ α [ I α ] = b is an atom, thereis a ∈ I α with b ≤ α ( a ) . By Lemma 2.13(1), α ∗ ( b ) ∈ ζ ( a ) . Thus, α ∗ [ X B ] ⊆ U . For thereverse inclusion, let y ∈ U . Then there is c ∈ I α with c ∈ y . Suppose ⋀ α ( y ) =
0. Then1 = ⋁ { ¬ α ( a ) ∣ a ∈ y } . Since y is a round filter, c ∈ y implies there is a ∈ y with a ≺ c ,so ¬ α ( ¬ a ) ≺ α ( c ) , and hence ¬ α ( c ) ≤ α ( ¬ a ) . Therefore, 1 = ⋁ { α ( ¬ a ) ∣ a ∈ y } , and so ¬ α ( ¬ c ) ≤ ⋁ { α ( ¬ a ) ∣ a ∈ y } . As c ∈ I α and y is closed under finite meets, there is a ∈ y with ¬ α ( ¬ c ) ≤ α ( ¬ a ) . Therefore, α ( a ) ≤ ¬ α ( ¬ a ) ≤ α ( ¬ c ) , yielding a ≤ ¬ c . Thus, a ∧ c =
0, which isfalse since a ∧ c ∈ y . Consequently, ⋀ α ( y ) ≠
0, and hence there is an atom b with b ≤ ⋀ α ( y ) .This implies that y ⊆ α − ( ↑ b ) , and since y is round, y ⊆ ↟ α − ( ↑ b ) = α ∗ ( b ) . Because y is anend, we have equality, and so y ∈ α ∗ [ X B ] . This completes the proof that α ∗ [ X B ] = U . (cid:3) From the claim we see that α ∗ [ X B ] is open in Y A , which implies that X B is locallycompact. (cid:3) Definition 4.5.
We call a de Vries extension α ∶ A → B locally compact provided α ( b ) = ⋁ { α ( a ) ∣ a ∈ I α , a ≺ b } for all b ∈ A . Remark 4.6.
By Theorem 2.4, we can phrase Theorem 4.3 dually as follows: Let e ∶ X → Y be a compactification and let α = e − ∶ RO ( Y ) → ℘ ( X ) be the corresponding de Vriesextension. Then the following are equivalent. (1) X is locally compact.(2) α is locally compact.(3) The ideal I α is round and ⋃ α [ I α ] = X .In particular, X is locally compact iff s − ∶ RO ( βX ) → ℘ ( X ) is a locally compact de Vriesextension.We next show that not every locally compact de Vries extension is maximal. Example 4.7.
Let X be the set of natural numbers equipped with the discrete topology,and let c ∶ X → ωX be the one-point compactification of X [4, Thm. 3.5.11]. By Remark 4.6,the de Vries extension c − ∶ RO ( Y ) → ℘ ( X ) is locally compact. However, since c is not iso-morphic to the Stone- ˇCech compactification of X , the de Vries extension c − is not maximalby [1, Thm. 6.4]. Definition 4.8. (1) Let
LDeVe be the full subcategory of
MDeVe consisting of locally compact maximalde Vries extensions.(2) Let
LKHaus be the full subcategory of
CReg consisting of locally compact spaces.
Theorem 4.9.
There is a dual equivalence between
LKHaus and
LDeVe .Proof.
Apply Theorems 2.9 and 4.3. (cid:3)
Another duality for
LKHaus was obtained by Dimov [3]. In Section 6 we will show how toderive Dimov’s duality from Theorem 4.9.5.
Compact de Vries extensions
In this section we introduce compact de Vries extensions and prove that the categoryof compact de Vries extensions is dually equivalent to the category of compact Hausdorffspaces. This yields that the category of compact de Vries extensions is equivalent to thecategory of de Vries algebras. We give a direct proof of this equivalence, thus providing adifferent perspective on de Vries duality.
Definition 5.1. (1) We call a de Vries extension α ∶ A → B compact provided the following axiom holds:If F is a round filter and I a round ideal of A with ⋀ α [ F ] ≤ ⋁ α [ I ] , then F ∩ I ≠ ∅ .(2) Let CDeVe be the full subcategory of
DeVe consisting of compact de Vries extensions.
Remark 5.2.
Every compact de Vries extension is normal. To see this, let α ∶ A → B be acompact de Vries extension, F a round filter, and I a round ideal of A with ⋀ α [ F ] ≤ ⋁ α [ I ] .Then there is a ∈ F ∩ I . Since I is round, there is b ∈ I with a ≺ b . Therefore, ⋀ α [ F ] ≤ α ( a ) and α ( b ) ≤ ⋁ α [ I ] . Thus, α is normal. Consequently, CDeVe is a full subcategory of
NDeVe ,and hence also of
MDeVe . Theorem 5.3.
For a de Vries extension α ∶ A → B , the following are equivalent. E VRIES DUALITY FOR NORMAL SPACES AND LOCALLY COMPACT HAUSDORFF SPACES 13 (1) α is a compact de Vries extension. (2) X B is compact. (3) I α = A .Proof. (1) ⇒ (2): Let x be an end of A . If ⋀ α [ x ] =
0, then for F = x and I = { } , (1) impliesthat 0 ∈ x , which is false. Therefore, there is an atom b of B with b ≤ ⋀ α [ x ] . This means x ⊆ α − ( ↑ b ) , and so x ⊆ ↟ α − ( ↑ b ) . Since ↟ α − ( ↑ b ) is a round filter and x is an end, we obtain x = ↟ α − ( ↑ b ) , and so x = α ∗ ( b ) . Thus, α ∗ [ X B ] = Y A , and hence X B is compact.(2) ⇒ (3): Suppose that X B is compact. Since α is isomorphic to α − ∗ ∶ RO ( Y A ) → ℘ ( X B ) and ¬ α − ∗ ( ¬ Y A ) = X B is compact, we see that 1 ∈ I α , so I α = A .(3) ⇒ (1): Suppose that I α = A . Let F be a round filter and I a round ideal of A with ⋀ α [ F ] ≤ ⋁ α [ I ] . Then ¬ ⋀ α [ F ] ∨ ⋁ α [ I ] =
1. We show that F ∩ I ≠ ∅ . Let J = { c ∈ A ∣ ¬ c ∈ F } . Since F is a round filter, J is a round ideal. We show that ¬ ⋀ α [ F ] = ⋁ α [ J ] . Since F is round, b ∈ F implies that there is a ∈ F with a ≺ b . Then α ( a ) ≤ ¬ α ( ¬ a ) ≤ α ( b ) . Therefore, ¬ ⋀ α [ F ] = ¬ ⋀ { ¬ α ( ¬ a ) ∣ a ∈ F } = ⋁ { α ( ¬ a ) ∣ a ∈ F } = ⋁ { α ( c ) ∣ c ∈ J } = ⋁ α [ J ] . From this equality we have 1 = ⋁ α [ J ] ∨ ⋁ α [ I ] . Since 1 ∈ I α , there are a , . . . , a n ∈ J and b , . . . , b m ∈ I with 1 = α ( a ) ∨ ⋯ ∨ α ( a n ) ∨ α ( b ) ∨ ⋯ ∨ α ( b m ) . Let a = a ∨ ⋯ ∨ a n and b = b ∨ ⋯ ∨ b m . Then a ∈ J , b ∈ I , α ( a ) ∨ ⋯ ∨ α ( a n ) ≤ α ( a ) , and α ( b ) ∨ ⋯ ∨ α ( b m ) ≤ α ( b ) .Therefore, 1 = α ( a ) ∨ α ( b ) , and so ¬ α ( ¬ a ) ∨ α ( b ) =
1, which gives α ( ¬ a ) ≤ α ( b ) . This implies ¬ a ≤ b , so ¬ a ∈ I . Because a ∈ J , we have ¬ a ∈ F , and hence ¬ a ∈ F ∩ I . Thus, α ∶ A → B is acompact de Vries extension. (cid:3) Remark 5.4.
By Theorem 2.4, we can phrase Theorem 5.3 dually as follows: Let e ∶ X → Y be a compactification and let α = e − ∶ RO ( Y ) → ℘ ( X ) be the corresponding de Vriesextension. Then the following are equivalent.(1) X is compact.(2) α is compact.(3) I α = RO ( Y ) .In particular, X is compact iff s − ∶ RO ( βX ) → ℘ ( X ) is a compact de Vries extension.Since KHaus is a full subcategory of
CReg , we may interpret it as a full subcategoryof
Comp . This interpretation sends a compact Hausdorff space X to the compactification X → X . Theorem 5.5.
CDeVe is dually equivalent to
KHaus .Proof.
By Remark 5.2,
CDeVe is a full subcategory of
MDeVe . The result then follows fromTheorems 2.9 and 5.3. (cid:3)
This together with de Vries duality yields that
CDeVe is equivalent
DeV . We give a directproof of this result, which provides a different perspective on de Vries duality.
Theorem 5.6.
CDeVe is equivalent to
DeV .Proof.
Define a functor D ∶ DeV → CDeVe as follows. If A is a de Vries algebra, then ζ A ∶ A → ℘ ( Y A ) is a de Vries extension, which is compact by Theorem 5.3, and we set D ( A ) = ζ A . To define D on morphisms, for a de Vries morphism ρ ∶ A → A ′ , consider thediagram A ζ A / / ρ (cid:15) (cid:15) ℘ ( Y A ) ρ − ∗ (cid:15) (cid:15) A ′ ζ A ′ / / ℘ ( Y A ′ ) . For a ∈ A , we have ρ − ∗ ( ζ A ( a )) = { y ∈ Y A ′ ∣ a ∈ ρ ∗ ( y )} = { y ∈ Y A ′ ∣ ∃ c ≺ a ∶ ρ ( c ) ∈ y } . Also, ( ζ A ′ ⋆ ρ )( a ) = ⋁ { ζ A ′ ( ρ ( c )) ∣ c ≺ a } . Since ζ A ′ ( ρ ( c )) = { y ∈ Y A ′ ∣ ρ ( c ) ∈ y } and ⋁ in ℘ ( Y A ′ ) is union, we see that ρ − ∗ ○ ζ A = ζ A ′ ⋆ ρ . Therefore, ( ρ, ρ − ∗ ) is a morphism in DeVe , and weset D ( ρ ) = ( ρ, ρ − ∗ ) . It is clear that D sends identity morphisms to identity morphisms. If ρ ∶ A → A ′ and τ ∶ A ′ → A ′′ are morphisms in DeV , then D ( τ ⋆ ρ ) = ( τ ⋆ ρ, ( τ ⋆ ρ ) − ∗ ) . Since ( τ ⋆ ρ ) − ∗ = ( ρ ∗ ○ τ ∗ ) − = τ − ∗ ○ ρ − ∗ , we see that D ( τ ⋆ ρ ) = ( τ ⋆ ρ, τ − ∗ ○ ρ − ∗ ) = ( τ, τ − ∗ ) ○ ( ρ, ρ − ∗ ) .Thus, D is a functor.Since D ( ρ ) = ( ρ, ρ − ∗ ) , it is clear that D is faithful. To see that D is full, let ( ρ, σ ) be amorphism between ζ A ∶ A → ℘ ( Y A ) and ζ A ′ ∶ A ′ → ℘ ( Y A ′ ) . Then σ ○ ζ A = ζ A ′ ⋆ ρ . If a ∈ A , thenas we saw above, σ ( ζ A ( a )) = ( ζ A ′ ⋆ ρ )( a ) = ρ − ∗ ( ζ A ( a )) . Therefore, since ζ A [ A ] is join-meetdense in ℘ ( Y A ) and σ, ρ − ∗ are both complete Boolean homomorphisms, we conclude that σ = ρ − ∗ . Thus, ( ρ, σ ) = ( ρ, ρ − ∗ ) = D ( ρ ) , and hence D is full.Let α ∶ A → B be a compact extension. Then X B is compact by Theorem 5.3, so α ∗ ∶ X B → Y A is a homeomorphism. Therefore, α is isomorphic to ζ A ∶ A → ℘ ( Y A ) . Thus, D ∶ DeV → CDeVe is an equivalence of categories [8, Thm. IV.4.1]. (cid:3) One-point Compactifications and minimal de Vries extensions
In this section we give an algebraic description of the one-point compactification of a non-compact locally compact Hausdorff space by introducing the concept of a minimal de Vriesextension.As we pointed out in Remark 2.8, a compactification e ∶ X → Y is equivalent to the Stone-ˇCech compactification s ∶ X → βX iff e is isomorphic to s in Comp . We next show thatif X is non-compact locally compact, then a corresponding result holds for the one-pointcompactification of X . Lemma 6.1.
Let X be a non-compact locally compact Hausdorff space and let e ∶ X → Y be a compactification. If e is isomorphic to the one-point compactification c ∶ X → ωX in Comp , then e and c are equivalent. E VRIES DUALITY FOR NORMAL SPACES AND LOCALLY COMPACT HAUSDORFF SPACES 15
Proof.
By hypothesis, there is an isomorphism ( f, g ) between e and c , which means thefollowing diagram is commutative. X e / / f (cid:15) (cid:15) Y g (cid:15) (cid:15) X c / / ωX Write ωX = c ( X ) ∪ { ∞ } . We claim that ∣ Y ∖ e [ X ]∣ =
1. Since g is onto, there is w ∈ Y with g ( w ) = ∞ . Then w ∉ e [ X ] since ∞ ∉ c [ X ] . Let y ∈ Y with y ≠ w . Then g ( y ) ≠ ∞ since g is 1-1. Therefore, there is x ∈ X with g ( y ) = c ( x ) . Since f is onto, there is x ′ ∈ X with x = f ( x ′ ) . Therefore, g ( y ) = c ( f ( x ′ )) = g ( e ( x ′ )) . Since g is 1-1, y = e ( x ′ ) . This proves Y ∖ e [ X ] = { w } , yielding the claim. Define h ∶ Y → ωX by h ( e ( x )) = c ( x ) if x ∈ X and h ( w ) = ∞ . This is well defined since e is 1-1. Note that h ○ e = c follows immediately fromthe definition. We prove that h is a homeomorphism. Let V be open in ωX . First supposethat ∞ ∉ V . Then V is open in c [ X ] , so V = c [ U ] for some open set U of X . As h is 1-1 and h ○ e = c , we see that that h − ( V ) = h − ( c [ U ]) = e [ U ] . This is open in Y since U is open in X and e [ X ] is open in Y . Next suppose that ∞ ∈ V . Then V = { ∞ } ∪ c [ U ] with U open in X and X ∖ U compact. By the previous case, we see that h − ( V ) = { w } ∪ e [ U ] . Furthermore, Y ∖ h − ( V ) = e [ X ] ∖ e [ U ] = e [ X ∖ U ] . Since X ∖ U is compact, e [ X ∖ U ] is compact, and so it is closed in Y . Thus, h − ( V ) isopen in Y . This completes the proof that h is continuous. Because it is a bijection betweencompact Hausdorff spaces, it is a homeomorphism. Thus, e ∶ X → Y and c ∶ X → ωX areequivalent as compactifications of X . (cid:3) Definition 6.2.
We say that a de Vries extension α ∶ A → B is minimal provided for everycompatible de Vries extension γ ∶ C → B , there is a de Vries morphism δ ∶ A → C such that γ ⋆ δ = α . A δ (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ α / / BC γ ? ? ⑦⑦⑦⑦⑦⑦⑦⑦ Theorem 6.3.
For a de Vries extension α ∶ A → B , the following are equivalent. (1) α is minimal but not compact. (2) X B is non-compact locally compact and α ∗ ∶ X B → Y A is isomorphic to the one-pointcompactification of X B . (3) X B is non-compact locally compact and α ∗ is equivalent to the one-point compactifi-cation of X B . (4) I α is an end ideal with ⋁ α [ I α ] = .Proof. (1) ⇒ (3). Since α is not compact, X B is non-compact by Theorem 5.3. Let e ∶ X B → Z be a compactification. Then e − ∶ RO ( Z ) → ℘ ( X B ) is a de Vries extension. As ζ B ∶ B → ℘ ( X B ) is a Boolean isomorphism, γ ∶ = ζ − B ○ e − ∶ RO ( Z ) → B is a de Vries extension, whichis compatible with α by Lemma 2.13(2). Since α is minimal, there is a de Vries morphism δ ∶ A → RO ( Z ) with γ ⋆ δ = α . By de Vries duality, δ ∗ induces a continuous map Z → Y A making the following diagram commute. X B e / / α ∗ ! ! ❈❈❈❈❈❈❈❈ Z (cid:127) (cid:127) ⑦⑦⑦⑦⑦⑦⑦⑦ Y A Therefore, α ∗ ∶ X B → Y A is the least compactification of X B . Thus, X B is locally compactand α ∗ ∶ X B → Y A is equivalent to the one-point compactification of X B by [4, Thm. 3.5.12]..(2) ⇔ (3). The implication ⇒ follows by Lemma 6.1 and the implication ⇐ is clear.(2) ⇒ (4). By Theorems 4.3 and 5.3, I α is a proper round ideal with ⋁ α [ I α ] =
1. Claim 4.4shows that the open subset of Y A corresponding to I α is α ∗ [ X B ] . Because this set is thecomplement of a single point, we see that I α is an end ideal.(4) ⇒ (2). Since I α is an end ideal, Claim 4.4 shows that Y A ∖ α ∗ [ X B ] is a single point.Therefore, X B is non-compact locally compact and α ∗ ∶ X B → Y A is isomorphic to theone-point compactification of X B .(3) ⇒ (1). By Theorem 5.3, α is not compact. Let γ ∶ C → B be a compatible extension.Then the topology on X B induced by γ is the same as that induced by α , and by [4,Thm. 3.5.11] there is a continuous map f ∶ Y C → Y A making the following diagram commute. X B γ ∗ / / α ∗ ! ! ❈❈❈❈❈❈❈❈ Y C f ~ ~ ⑥⑥⑥⑥⑥⑥⑥ Y A By de Vries duality, there is a de Vries morphism δ ∶ A → C with δ ∗ = f . Since the functor E ∶ Comp → DeVe is faithful, γ ⋆ δ = α , and hence α is a minimal de Vries extension. (cid:3) Remark 6.4.
Using Theorem 2.4, we can phrase Theorem 6.3 as follows: Let e ∶ X → Y bea compactification and let α = e − ∶ RO ( Y ) → ℘ ( X ) be the corresponding de Vries extension.Then the following are equivalent.(1) X is non-compact locally compact and e ∶ X → Y is isomorphic to the one-pointcompactification of X .(2) X is non-compact locally compact and e ∶ X → Y is equivalent to the one-pointcompactification of X .(3) α is minimal but not compact.(4) I α is an end ideal with ⋃ α [ I α ] = X . Remark 6.5.
Let α ∶ A → B be a de Vries extension.(1) If α ∶ A → B is compact, then every compatible de Vries extension is isomorphic to α .To see this, let γ ∶ C → B be compatible with α . By Lemma 2.13(2), the topology on E VRIES DUALITY FOR NORMAL SPACES AND LOCALLY COMPACT HAUSDORFF SPACES 17 X B inherited from α is the same as that inherited from γ . Because α is compact, thespace X B is compact by Theorem 5.3. Consequently, the embeddings α ∗ ∶ X B → Y A and γ ∗ ∶ X B → Y C are both homeomorphisms, and so α ∗ and γ ∗ are isomorphic in Comp as we see from the following diagram. X B α ∗ / / Y A γ ∗ ○ α − ∗ (cid:15) (cid:15) X B γ ∗ / / Y C By Theorem 2.4, it follows that α and γ are isomorphic.(2) We show that α is both maximal and minimal iff X B is almost compact, where werecall (see, e.g., [5, p. 95]) that a completely regular space X is almost compact provided ∣ βX ∖ X ∣ ≤
1. First suppose that α is not compact. By Theorem 5.3, X B isnon-compact. Therefore, by [1, Thm. 6.4] and Theorem 6.3, α is both maximal andminimal iff X B is almost compact. Next suppose that α is compact. Then X B iscompact. Also, by (1), each compatible de Vries extension is isomorphic to α , whichimplies that α is both maximal and minimal. Consequently, α is both maximal andminimal iff X B is almost compact.7. Dimov duality for
LKHaus
In Theorem 4.9 we proved that
LKHaus is dually equivalent to
LDeVe . In [3, Thm. 3.12]Dimov proved that
LKHaus is dually equivalent to a category we denote by
Dim below. Itfollows that there is an equivalence between
LDeVe and
Dim . LKHaus y y ttttttttt f f & & ▲▲▲▲▲▲▲▲▲▲ Dim o o / / LDeVe
In this section we give a direct proof for why
LDeVe and
Dim are equivalent, thus obtainingDimov duality as a consequence of Theorem 4.9.
Definition 7.1. A Dimov algebra is a triple D = ( A, ⊲ , I ) , where A is a complete Booleanalgebra, ⊲ is a binary relation on A satisfying (DV1)–(DV5) of Definition 2.1, and I is anideal of A satisfying(I1) If a ∈ I and a ⊲ b , then there is c ∈ I with a ⊲ c ⊲ b .(I2) If ( a ∧ c ) ⊲ ( b ∨ ¬ c ) for all c ∈ I , then a ⊲ b .(I3) If b ≠
0, then there is 0 ≠ a ∈ I with a ⊲ b . Remark 7.2.
In [3] Dimov worked with contact relations δ and the resulting contact alge-bras. The two relations δ and ⊲ are dual to each other: aδb iff a / ⊲ ¬ b . The axioms in termsof δ are given in [3, Def. 2.1], and it is straightforward to see that they are equivalent toaxioms (DV1)-(DV5) for ⊲ . Axioms (I1) and (I3) are the same as the axioms Dimov gives in [3, Def. 2.9]. Our axiom (I2) is slightly different from the corresponding axiom of Dimov,which can be phrased in the language of ⊲ as follows:( ∗ ) If a ⊲ ( b ∨ ¬ c ) for all c ∈ I, then a ⊲ b. Clearly (I2) implies ( ∗ ) . For the converse, suppose that ( ∗ ) holds and ( a ∧ c ) ⊲ ( b ∨ ¬ c ) forall c ∈ I . Let d ∈ I . For each c ∈ I , we have a ∧ d ≤ ( a ∧ d ) ∨ ( a ∧ c ) = a ∧ ( c ∨ d ) . Since c ∨ d ∈ I ,by assumption, a ∧ ( c ∨ d ) ⊲ b ∨ ¬ ( c ∨ d ) . Since b ∨ ¬ ( c ∨ d ) ≤ b ∨ ¬ c , we have a ∧ d ⊲ b ∨ ¬ c forall c ∈ I by (DV3). Thus, ( ∗ ) implies ( a ∧ d ) ⊲ b . Since this is true for all d ∈ I , by (DV5), ¬ b ⊲ ( ¬ a ∨ ¬ d ) for all d ∈ I . Applying ( ∗ ) again yields ¬ b ⊲ ¬ a , so a ⊲ b by (DV5). Thisshows that we can replace ( ∗ ) by (I2) above. Definition 7.3.
A map ρ ∶ D → D ′ between Dimov algebras is a Dimov morphism if ρ satisfies axioms (M1) and (M2) of Definition 2.1 together with(D3) If a ⊲ b , then ¬ ρ ( ¬ a ) ⊲ ρ ( b ) .(D4) If c ∈ I ′ , then there is a ∈ I with c ≤ ρ ( a ) .(D5) ρ ( b ) = ⋁ { ρ ( a ) ∣ a ∈ I, a ⊲ b } .Similar to the de Vries setting, the composition of two Dimov morphisms ρ and ρ isgiven by ( ρ ◇ ρ )( b ) = ⋁ { ρ ( ρ ( a )) ∣ a ∈ I, a ⊲ b } . Like
DeV , with this composition, Dimov algebras and Dimov morphisms form a category(see [3, Prop. 4.24]), which we denote by
Dim . Remark 7.4.
In Dimov’s original definition [3, Def. 3.8], a weaker version of Axiom (D3) isused:( ∗∗ ) If a ∈ I and a ⊲ b, then ¬ ρ ( ¬ a ) ⊲ ρ ( b ) . But it follows from [3, Lem 4.19] that the two axioms are equivalent. Dimov’s proof isn’tpoint-free; we give an alternative, pointfree proof. Suppose ( ∗∗ ) holds. To see that (D3)holds, let a ⊲ b . To show that ¬ ρ ( ¬ a ) ⊲ ρ ( b ) , by (I2) it is sufficient to prove that ¬ ρ ( ¬ a ) ∧ c ⊲ ρ ( b ) ∨ ¬ c for each c ∈ I ′ . Let c ∈ I ′ . By (D4), there is d ∈ I with c ≤ ρ ( d ) , and so ¬ ρ ( ¬ a ) ∧ c ≤ ¬ ρ ( ¬ a ) ∧ ρ ( d ) . Claim 7.5. If ρ ∶ A → A ′ is a meet preserving function between Boolean algebras, then ¬ ρ ( ¬ a ) ∧ ρ ( d ) ≤ ¬ ρ ( ¬ ( a ∧ d )) for all a, d ∈ A .Proof of the Claim . We have d ∧ ¬ a ≤ ¬ a ⇒ ρ ( d ∧ ¬ a ) ≤ ρ ( ¬ a ) ⇒ ρ ( d ∧ ¬ ( a ∧ d )) ≤ ρ ( ¬ a ) ⇒ ρ ( d ) ∧ ρ ( ¬ ( a ∧ d )) ≤ ρ ( ¬ a ) ⇒ ¬ ρ ( ¬ a ) ∧ ρ ( d ) ∧ ρ ( ¬ ( a ∧ d )) = ⇒ ¬ ρ ( ¬ a ) ∧ ρ ( d ) ≤ ¬ ρ ( ¬ ( a ∧ d )) . Thus, the claim holds. (cid:3)
E VRIES DUALITY FOR NORMAL SPACES AND LOCALLY COMPACT HAUSDORFF SPACES 19
By Claim 7.5, ¬ ρ ( ¬ a ) ∧ ρ ( d ) ≤ ¬ ρ ( ¬ ( a ∧ d )) . But since d ∈ I , we have a ∧ d ∈ I , so ( ∗∗ )yields ¬ ρ ( ¬ ( a ∧ d )) ⊲ ρ ( b ) . So ¬ ρ ( ¬ a ) ∧ c ≤ ¬ ρ ( ¬ a ) ∧ ρ ( d ) ≤ ¬ ρ ( ¬ ( a ∧ d )) ⊲ ρ ( b ) ≤ ρ ( b ) ∨ ¬ c for each c ∈ I ′ . Thus, ¬ ρ ( ¬ a ) ⊲ ρ ( b ) . Lemma 7.6.
Suppose that α ∶ A → B is a locally compact de Vries extension. Define ⊲ on A by a ⊲ b iff ( a ∧ c ) ≺ ( b ∨ ¬ c ) for all c ∈ I α . (1) If a ≺ b , then a ⊲ b . (2) If a ⊲ b , then ¬ α ( ¬ a ) ≤ α ( b ) . (3) If a ∈ I α and a ⊲ b , then a ≺ b .Proof. (1). If c ∈ I α , then ( a ∧ c ) ≤ a ≺ b ≤ b ∨ ¬ c , so ( a ∧ c ) ≺ ( b ∨ ¬ c ) , and hence a ⊲ b .(2). For each c ∈ I α we have ( a ∧ c ) ≺ ( b ∨ ¬ c ) , so ¬ α ( ¬ ( a ∧ c )) ≤ α ( b ∨ ¬ c ) . By Claim 7.5,we have ¬ α ( ¬ a ) ∧ α ( c ) ≤ ¬ α ( ¬ ( a ∧ c )) . We show that α ( b ∨ ¬ c ) ≤ α ( b ) ∨ ¬ α ( c ) . To see this, b ∧ c ≤ b , so we obtain: b ∧ c ≤ b ⇒ α ( b ∧ c ) ≤ α ( b ) ⇒ α (( b ∨ ¬ c ) ∧ c ) ≤ α ( b ) ⇒ α ( b ∨ ¬ c ) ∧ α ( c ) ≤ α ( b ) ⇒ α ( b ∨ ¬ c ) ≤ α ( b ) ∨ ¬ α ( c ) . Therefore, ¬ α ( ¬ a ) ∧ α ( c ) ≤ α ( b ) ∨ ¬ α ( c ) . Thus, ¬ α ( ¬ a ) ∧ ¬ α ( b ) ∧ α ( c ) = c ∈ I α . Since α is locally compact, ⋁ { α ( c ) ∣ c ∈ I α } = = ⋁ { ¬ α ( ¬ a ) ∧ ¬ α ( b ) ∧ α ( c ) ∣ c ∈ I α } = ¬ α ( ¬ a ) ∧ ¬ α ( b ) ∧ ⋁ { α ( c ) ∣ c ∈ I α } = ¬ α ( ¬ a ) ∧ ¬ α ( b ) . Therefore, ¬ α ( ¬ a ) ∧ ¬ α ( b ) =
0, and so ¬ α ( ¬ a ) ≤ α ( b ) .(3). By (2), ¬ α ( ¬ a ) ≤ α ( b ) . Since α is locally compact, α ( b ) = ⋁ { α ( c ) ∣ c ∈ I α , c ≺ b } .Because a is α -compact, applying Lemma 2.11 to the ideal I α ∩↡ b yields c ∈ I α with ¬ α ( ¬ a ) ≤ α ( c ) and c ≺ b . As α ( a ) ≤ ¬ α ( ¬ a ) and α is an embedding, we have a ≤ c , so a ≺ b . (cid:3) We define D ∶ LDeVe → Dim as follows. For a locally compact de Vries extension α ∶ A → B ,let D ( α ) = ( A, ⊲ , I α ) , where A is the underlying complete Boolean algebra of A and ⊲ isdefined by a ⊲ b iff ( a ∧ c ) ≺ ( b ∨ ¬ c ) for all c ∈ I α as in the statement of Lemma 7.6; and if ( ρ, σ ) is a morphism in LDeVe , let D ( ρ, σ ) = ρ . Proposition 7.7. D ∶ LDeVe → Dim is a covariant functor.Proof.
Let α ∶ A → B be a locally compact de Vries extension. We show that D ( α ) ∈ Dim .For this, we first show that ⊲ satisfies (DV1)-(DV5).(DV1) is clear. (DV2): If a ⊲ b , then ( a ∧ c ) ≺ ( b ∨ ¬ c ) for all c ∈ I α . This implies ( a ∧ c ) ≤ ( b ∨ ¬ c ) , andso a ∧ ¬ b ∧ c = c ∈ I α . Since α is locally compact, ⋁ α [ I α ] =
1. Because α is anembedding, the last equality implies ⋁ I α =
1. Therefore, a ∧ ¬ b =
0, so a ≤ b .(DV3): If a ≤ b ⊲ c ≤ d , then for all e ∈ I α we have a ∧ e ≤ b ∧ e ≺ c ∨ ¬ e ≤ d ∨ ¬ e . Thus, a ⊲ d .(DV4): Suppose that a ⊲ b, c . Then ( a ∧ e ) ≺ ( b ∨ ¬ e ) , ( c ∨ ¬ e ) for all e ∈ I α . Thus, ( a ∧ e ) ≺ (( b ∧ c ) ∨ ¬ e ) for all e ∈ I α , so a ⊲ ( b ∧ c ) .(DV5): If a ⊲ b , then ( a ∧ c ) ≺ ( b ∨ ¬ c ) for all c ∈ I α . Therefore, ( ¬ b ∧ c ) ≺ ( ¬ a ∨ ¬ c ) for all c ∈ I α . Thus, ¬ b ⊲ ¬ a .We next show that I α satisfies (I1)-(I3).(I1): Let a ∈ I α and a ⊲ b . Then ¬ α ( ¬ a ) ≤ α ( b ) by Lemma 7.6(2). Since α is locally com-pact, we have α ( b ) = ⋁ { α ( c ) ∣ c ∈ I α , c ≺ b } . Because a is α -compact, applying Lemma 2.11 to I α ∩ ↡ b yields c ∈ I α with c ≺ b and ¬ α ( ¬ a ) ≤ α ( c ) . Repeating this argument with b replacedby c yields d ∈ I α with d ≺ c and ¬ α ( ¬ a ) ≤ α ( d ) . Since α ( a ) ≤ ¬ α ( ¬ a ) , we see that a ≤ d ≺ c ,so a ≺ c ≺ b . Thus, a ⊲ c ⊲ b by Lemma 7.6(1).(I2): If ( a ∧ c ) ⊲ ( b ∨ ¬ c ) for all c ∈ I α , then a ∧ c ∈ I α , so ( a ∧ c ) ≺ ( b ∨ ¬ c ) for all c ∈ I α byLemma 7.6(3). Thus, a ⊲ b .(I3): Suppose b ≠
0. Then α ( b ) ≠ α is an embedding. Therefore, as α ( b ) = ⋁ { α ( a ) ∣ a ∈ I α , a ≺ b } , there is 0 ≠ a ∈ I α with a ≺ b . By Lemma 7.6(1), a ⊲ b .This shows that D ( α ) ∈ Dim , and hence D is well defined on objects. To see that D is welldefined on morphisms, let ( ρ, σ ) be a morphism between locally compact de Vries extensions α ∶ A → B and α ′ ∶ A ′ → B ′ . A α / / ρ (cid:15) (cid:15) B σ (cid:15) (cid:15) A ′ α ′ / / B ′ We show that ρ ∶ D ( α ) → D ( α ′ ) is a morphism in Dim . Since ρ is a de Vries morphism,it satisfies (M1) and (M2). Suppose a ∈ I α with a ⊲ b . Then a ≺ b by Lemma 7.6(3), so ¬ ρ ( ¬ a ) ≺ ρ ( b ) . This implies ¬ ρ ( ¬ a ) ⊲ ρ ( b ) by Lemma 7.6(1), so (D3) holds.To verify (D4) we point out that if b ∈ I α , then σ ( α ( b )) = ( α ′ ⋆ ρ )( b ) = ⋁ { α ′ ( ρ ( a )) ∣ a ∈ I α , a ≺ b } . Therefore, ⋁ { σ ( α ( b )) ∣ b ∈ I α } = ⋁ { α ′ ( ρ ( a )) ∣ a ∈ I α } . But ⋁ { σ ( α ( b )) ∣ b ∈ I α } = σ ( ⋁ { α ( b ) ∣ b ∈ I α }) = σ ( ) = . So, if c ∈ I α ′ , then ¬ α ′ ( ¬ c ) ≤ ⋁ { α ′ ( ρ ( a )) ∣ a ∈ I α } . Since c is α ′ -compact and I α is an ideal,there is a ∈ I α with ¬ α ′ ( ¬ c ) ≤ α ′ ( ρ ( a )) . As α ′ is an embedding and α ′ ( c ) ≤ ¬ α ′ ( ¬ c ) , weconclude that c ≤ ρ ( a ) . Thus, (D4) holds.For (D5), since ⋁ I α ′ =
1, (D4) implies that 1 = ⋁ { ρ ( a ) ∣ a ∈ I α } . Let b ∈ A . Then ρ ( b ) = ρ ( b ) ∧ ⋁ { ρ ( a ) ∣ a ∈ I α } = ⋁ { ρ ( a ∧ b ) ∣ a ∈ I α } = ⋁ { ρ ( c ) ∣ c ∈ I α , c ≤ b } . E VRIES DUALITY FOR NORMAL SPACES AND LOCALLY COMPACT HAUSDORFF SPACES 21
Now, if c ∈ I α with c ≤ b , then ρ ( c ) = ⋁ { ρ ( a ) ∣ a ≺ c } by (M4). For a with a ≺ c , we have a ∈ I α and a ⊲ c by Lemma 7.6(1). Thus, ρ ( b ) = ⋁ { ρ ( a ) ∣ a ∈ I α , a ⊲ b } , and so (D5) holds.It follows that D ( ρ, σ ) = ρ is a morphism in Dim . It is clear that D preserves identity maps.To see that D preserves composition, let ( ρ , σ ) and ( ρ , σ ) be composable morphisms in LDeVe . Their composition is ( ρ ⋆ ρ , σ ○ σ ) . We have shown above that ρ ⋆ ρ is then amorphism of Dim . Therefore, by (D5), ( ρ ⋆ ρ )( b ) = ⋁ {( ρ ⋆ ρ )( a ) ∣ a ∈ I α , a ≺ b } . On the other hand, by the definition of the composition ρ ◇ ρ in Dim , ( ρ ◇ ρ )( b ) = ⋁ { ρ ( ρ ( a )) ∣ a ∈ I α , a ⊲ b } . By Lemma 7.6, for a ∈ I α , we have a ≺ b iff a ⊲ b . Therefore, since ( ρ ⋆ ρ )( a ) ≤ ρ ( ρ ( a )) we get ( ρ ⋆ ρ )( b ) ≤ ( ρ ◇ ρ )( b ) . However, by definition of ⋆ , ( ρ ⋆ ρ )( b ) = ⋁ {( ρ ( ρ ( a )) ∣ a ≺ b } , which gives the reverse inequality. Therefore, ρ ⋆ ρ = ρ ◇ ρ , which shows that D preservescomposition. Thus, D is a covariant functor. (cid:3) Our goal is to see that D is an equivalence. For this we need to produce, for a Dimovalgebra D , a maximal locally compact de Vries extension. Let D = ( A, ⊲ , I ) be a Dimovalgebra. The construction in the following definition is well known in pointfree topology(see, e.g., [6, p. 126] or [10, p. 90]). Definition 7.8.
We define ≺ on D by a ≺ b iff there is a family { c p ∣ p ∈ Q ∩ [ , ]} with a ≤ c , c ≤ b , and c p ⊲ c q for each p < q . We call the sequence { c p } an interpolating sequencewitnessing a ≺ b .Recall (see, e.g., [10, p. 90]) that a binary relation R is said to be interpolating if aRb implies there is c with aRc and cRb . It is standard to see that ≺ is the largest interpolatingrelation contained in ⊲ . Remark 7.9.
Suppose that D is a Dimov algebra and ≺ is given as in Definition 7.8. If a ⊲ b and a ∈ I , then repeated use of (I1) shows that a ≺ b .In order to prove Theorem 7.11, we require the following characterization of compactifi-cations of a completely regular space [2, Thm. 2.2.4], which is de Vries’ pointfree versionof Smirnov’s theorem. If X is a completely regular space, define ⊲ on RO ( X ) by U ⊲ V if cl ( U ) ⊆ V . If ≺ is a proximity on RO ( X ) , we say that ≺ is compatible with the topology if ≺ is contained in ⊲ and V = ⋃ { U ∈ RO ( X ) ∣ ∃ W ∈ RO ( X ) , U ≺ W ⊆ V } for each open set V . Theorem 7.10 (de Vries) . Let X be a completely regular space. There is an order isomor-phism between the poset of (inequivalent) compactifications of X and the poset of proximities ≺ on RO ( X ) compatible with the topology. Theorem 7.11.
Let D = ( A, ⊲ , I ) be a Dimov algebra. The relation ≺ defined in Defini-tion 7.8 is a de Vries proximity and I is a round ideal of the de Vries algebra A ∶ = ( A, ≺ ) .Moreover, if X is the open subspace of Y A corresponding to I , then X is locally compactand dense in Y A , and the inclusion map e ∶ X → Y A is isomorphic to the Stone- ˇCech com-pactification of X . Furthermore, if α ∶ A → ℘ ( X ) is the locally compact de Vries extensioncorresponding to e , then I = I α .Proof. We first show that ≺ is a de Vries proximity.(DV1). The constant sequence { } is an interpolating sequence, so 1 ≺ a ≺ b and { c p } is an interpolating sequence, then c ⊲ c , so c ≤ c . Thus, a ≤ b .(DV3). If a ≤ b ≺ c ≤ d and { e p } is an interpolating sequence witnessing b ≺ c , then it isclear that { e p } is also an interpolating sequence witnessing a ≺ d .(DV4). Let a ≺ b, c and let { d p } , { e p } be interpolating sequences witnessing a ≺ b and a ≺ c ,respectively. Set f p = d p ∧ e q . Then { f p } is an interpolating sequence witnessing a ≺ ( b ∧ c ) .(DV5). Let a ≺ b and { c p } be an interpolating sequence. Set d p = ¬ c − p . Then a ≤ c and c ≤ b yield ¬ b ≤ d and d ≤ ¬ a . Moreover, if p < q , then 1 − q < − p , so c − q ⊲ c − p . Therefore, d p = ¬ c − p ⊲ ¬ c − q = d q . Thus, { d q } is an interpolating sequence witnessing ¬ b ≺ ¬ a .(DV6). Let a ≺ b and let { c p } be an interpolating sequence. Set c = c / . If d p = c p / and e p = c ( + p )/ , then it is well known and straightforward to see that { d p } is an interpolatingsequence witnessing a ≺ c and { e p } is an interpolating sequence witnessing c ≺ b .(DV7). If b ≠
0, then there is 0 ≠ a ∈ I with a ⊲ b . Thus, a ≺ b by a repeated use of (I1).This proves that A is a de Vries algebra. We next show I is a round ideal of A . Let a ∈ I .Since a ⊲
1, by (I1) there is b ∈ I with a ⊲ b ⊲
1. Therefore, a ≺ b . This shows that I is round.In addition, ⋁ I = ¬ ⋁ I ≠
0, so by (I3), there is 0 ≠ a ∈ I with a ⊲ ¬ ⋁ I .Thus, a ≤ ⋁ I, ¬ ⋁ I , yielding a =
0. The obtained contradiction proves that ⋁ I = I is a round ideal, by Lemma 2.10, I corresponds to the open subset X ∶ = ⋃ { ζ ( a ) ∣ a ∈ I } of Y A . As ⋁ I =
1, from ζ ( ⋁ I ) = int Y A ( cl Y A ( X )) it follows that X is dense in Y A .Since X is an open subset of Y A , it is locally compact; and since X is dense, the inclusionmap e ∶ X → Y A is a compactification of X . Consider the locally compact de Vries extension α ∶ A → ℘ ( X ) corresponding to e ∶ X → Y A and given by α ( a ) = ζ ( a ) ∩ X . We show that I = I α . Let a ∈ I . Since X = ⋃ { ζ ( a ) ∣ a ∈ I } , we have ζ ( a ) ⊆ X . Because I is round,this implies that cl Y A ( ζ ( a )) ⊆ X . But cl Y A ( ζ ( a )) = Y A ∖ ζ ( ¬ a ) . So Y A ∖ ζ ( ¬ a ) ⊆ X , Thus, ¬ α ( ¬ a ) = X ∖ α ( ¬ a ) = X ∖ ζ ( ¬ a ) . Because Y A ∖ ζ ( ¬ a ) ⊆ X , we see that ¬ α ( ¬ a ) = Y A ∖ ζ ( ¬ a ) = cl Y A ( ζ ( a )) . Since cl Y A ( ζ ( a )) is a compact subset of X , we conclude that a ∈ I α .Conversely, let a ∈ I α . Then X ∖ α ( ¬ a ) is a compact subset of X . Since X = ⋃ { ζ ( b ) ∣ b ∈ I } , we have α ( a ) ⊆ ¬ α ( ¬ a ) = X ∖ α ( ¬ a ) ⊆ ⋃ { ζ ( b ) ∣ b ∈ I } . As X ∖ α ( ¬ a ) is compactand I is an ideal, there is b ∈ I with α ( a ) ⊆ ζ ( b ) . Because X is dense in Y A , we have cl Y A ( ζ ( a )) = cl Y A ( ζ ( a ) ∩ X ) = cl Y A ( α ( a )) ⊆ cl Y A ( ζ ( b )) . Since I is a round ideal, there is c ∈ I with cl Y A ( ζ ( b )) ⊆ ζ ( c ) . Therefore, ζ ( a ) ⊆ ζ ( c ) , so a ≤ c , and hence a ∈ I . Thus, I = I α . E VRIES DUALITY FOR NORMAL SPACES AND LOCALLY COMPACT HAUSDORFF SPACES 23
We finish the proof by showing that e is the Stone- ˇCech compactification of X . By a dualargument to that in Remark 7.2, we have a ⊲ b iff a ∧ c ⊲ b for all c ∈ I . If a ∈ I , then a ⊲ b implies a ≺ b by Remark 7.9. In the following, we will identify A with RO ( Y A ) and write Y = Y A . We have U ⊲ V ← → U ∩ W ⊲ V for all W ∈ I ← → U ∩ W ≺ V for all W ∈ I ← → cl Y ( U ∩ W ) ⊆ V for all W ∈ I ← → ⋃ { cl Y ( U ∩ W ) ∣ W ∈ I } ⊆ V. We claim that ⋃ { cl Y ( U ∩ W ) ∣ W ∈ I } = cl X ( U ∩ X ) . To see this we first recall that since I is round and ⋃ I = X , if W ∈ I , there is W ′ ∈ I with W ≺ W ′ , and so cl Y ( W ) ⊆ W ′ ⊆ X .Therefore, cl Y ( W ) is a closed subset of X , and so cl Y ( W ) = cl X ( W ) . From this observationthe inclusion ⊆ is clear. For the reverse inclusion, let x ∈ cl X ( U ∩ X ) . By definition of X ,there is W ∈ I with x ∈ W . If V is an open neighborhood of x , then as x ∈ cl X ( U ∩ X ) and V ∩ W is an open neighborhood of x , we have U ∩ X ∩ ( V ∩ W ) ≠ ∅ , so ( U ∩ W ) ∩ V ≠ ∅ .Thus, x ∈ cl Y ( U ∩ W ) . This verifies the claim. We have therefore shown that U ⊲ V iff cl X ( U ∩ X ) ⊆ V ∩ X .By Lemma 2.12, the map U ↦ U ∩ X is a Boolean isomorphism from RO ( Y ) to RO ( X ) .This allows us to move ⊲ to the relation on RO ( X ) , which we also denote by ⊲ , given by U ⊲ V iff cl X ( U ) ⊆ V . Similarly, we can move ≺ to a proximity on RO ( X ) . By a standardUrysohn argument, ≺ is the largest proximity on RO ( X ) contained in ⊲ . By Theorem 7.10,the proximity ≺ corresponds to the largest compactification of X , and so e ∶ X → Y A is theStone- ˇCech compactification of X . This completes the proof. (cid:3) Theorem 7.12.
The categories
LDeVe and
Dim are equivalent.Proof.
By Proposition 7.7, we have a covariant functor D ∶ LDeVe → Dim . By [8, Thm. IV.4.1],it is sufficient to show that each object of
Dim is isomorphic to the D -image of an object of LDeVe , and that D is full and faithful.Let D = ( A, ⊲ , I ) ∈ Dim . By Theorem 7.11, we have a locally compact extension α ∶ A →℘ ( X ) with I = I α . The functor D sends this extension to ( A, ⊲ ′ , I ) , where ⊲ ′ is defined inthe statement of Lemma 7.6. We show that ⊲ ′ = ⊲ . If a ⊲ b , then a ∧ c ≤ a ⊲ b ≤ b ∨ ¬ c , so ( a ∧ c ) ⊲ ( b ∨ ¬ c ) for each c ∈ I . Since a ∧ c ∈ I , we have ( a ∧ c ) ≺ ( b ∨ ¬ c ) by Remark 7.9.Thus, a ⊲ ′ b . Conversely, suppose that a ⊲ ′ b . Then ( a ∧ c ) ≺ ( b ∨ ¬ c ) for all c ∈ I . Therefore, ( a ∧ c ) ⊲ ( b ∨ ¬ c ) by the definition of ≺ . Thus, a ⊲ b by (I2). This proves that D = D ( α ) .To see that D is faithful, since D ( ρ, σ ) = ρ for each morphism ( ρ, σ ) of LDeVe , it sufficesto show that if ( ρ, σ ) and ( ρ, ψ ) are morphisms between α ∶ A → B and α ′ ∶ A ′ → B ′ , then σ = ψ . A ρ (cid:15) (cid:15) α / / B σ (cid:10) (cid:10) ψ (cid:20) (cid:20) A ′ α ′ / / B ′ We have σ ○ α = ψ ○ α since both are equal to α ′ ⋆ ρ . Both σ and ψ are complete Booleanhomomorphisms. Since α [ A ] is join-meet dense in B , it follows that σ = ψ . This shows that D is faithful.Finally, to see that D is full, for each morphism ρ ∶ D → D ′ in Dim , we need to produce amorphism ( ρ, σ ) in LDeVe . Using the construction of Theorem 7.11, we have locally compactde Vries extensions α ∶ A → ℘ ( X ) and α ′ ∶ A ′ → ℘ ( X ′ ) with I = I α and I ′ = I α ′ .We first show that ρ ∶ A → A ′ is a de Vries morphism. Clearly (M1) and (M2) hold. Toprove (M3), suppose that a ≺ b . Then there is an interpolating sequence { c p } witnessing a ≺ b .For each p < q , since c p ⊲ c q , we have ¬ ρ ( ¬ c p ) ⊲ ρ ( c q ) by (D3). Set d = ¬ ρ ( ¬ c ) , and d p = ρ ( c p ) if p >
0. Then p < q implies d p ⊲ d q as d p ≤ ¬ ρ ( ¬ c p ) . Moreover, ¬ ρ ( ¬ a ) ≤ ¬ ρ ( ¬ c ) = d .Consequently, ¬ ρ ( ¬ a ) ≺ ρ ( b ) . Finally, for (M4), let b ∈ A . Then ρ ( b ) = ⋁ { ρ ( a ) ∣ a ∈ I, a ⊲ b } .However, if a ∈ I and a ⊲ b , then a ≺ b by Remark 7.9. Therefore, ρ ( b ) = ⋁ { ρ ( a ) ∣ a ≺ b } , andso (M4) holds. Thus, ρ is a de Vries morphism.We next show that ρ ∗ ( X ′ ) ⊆ X . If x ∈ X ′ , then there is b ∈ I ′ with x ∈ ζ ( b ) ⊆ X ′ . By(D4), there is a ∈ I with b ≤ ρ ( a ) . Since I is round, there is c ∈ I with a ≺ c . We have b ∈ x ,so ρ ( a ) ∈ x . Therefore, a ∈ ρ − ( x ) , and so c ∈ ↟ ρ − ( x ) = ρ ∗ ( x ) . Thus, ρ ∗ ( x ) ∈ ζ ( c ) ⊆ X , asdesired. The restriction of ρ ∗ to X ′ is then a well defined function X ′ → X , and so there isa complete Boolean homomorphism σ ∶ ℘ ( X ) → ℘ ( X ′ ) given by σ ( S ) = ( ρ ∗ ) − ( S ) for each S ⊆ X . A α / / ρ (cid:15) (cid:15) ℘ ( X ) σ (cid:15) (cid:15) A ′ α ′ / / ℘ ( X ′ ) To see that ( ρ, σ ) is a morphism in LDeVe , we must show that σ ○ α = α ′ ⋆ ρ . Let b ∈ A . Then σ ( α ( b )) = σ ( ζ ( b ) ∩ X ) = ( ρ ∗ ) − ( ζ ( b ) ∩ X ) = { x ∈ X ′ ∣ b ∈ ρ ∗ ( x )} = { x ∈ X ′ ∣ ∃ a ≺ b ∶ ρ ( a ) ∈ x } . On the other hand, ( α ′ ⋆ ρ )( b ) = ⋁ { α ′ ( ρ ( a )) ∣ a ≺ b } . Now, for a ≺ b , we have α ′ ( ρ ( a )) = ζ ( ρ ( a )) ∩ X ′ = { x ∈ X ′ ∣ ρ ( a ) ∈ x } . Therefore, as the join in ℘ ( X ′ ) is union, ( α ′ ⋆ ρ )( b ) = { x ∈ X ′ ∣ ∃ a ≺ b ∶ ρ ( a ) ∈ x } = σ ( α ( b )) . E VRIES DUALITY FOR NORMAL SPACES AND LOCALLY COMPACT HAUSDORFF SPACES 25
This shows that ( ρ, σ ) is a morphism in LDeVe . Since D ( ρ, σ ) = ρ , we conclude that D isfull. This completes the proof that D is part of a category equivalence between LDeVe and
Dim . (cid:3) References [1] G. Bezhanishvili, P. J. Morandi, and B. Olberding,
De Vries duality for compactifications and completelyregular spaces , Submitted. Preprint available at arXiv:1804.03210, 2018.[2] H. de Vries,
Compact spaces and compactifications. An algebraic approach , Ph.D. thesis, University ofAmsterdam, 1962.[3] G. D. Dimov,
A de Vries-type duality theorem for the category of locally compact spaces and continuousmaps. I , Acta Math. Hungar. (2010), no. 4, 314–349.[4] R. Engelking,
General topology , second ed., Sigma Series in Pure Mathematics, vol. 6, HeldermannVerlag, Berlin, 1989.[5] L. Gillman and M. Jerison,
Rings of continuous functions , The University Series in Higher Mathematics,D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960.[6] P. T. Johnstone,
Stone spaces , Cambridge Studies in Advanced Mathematics, vol. 3, Cambridge Uni-versity Press, Cambridge, 1982.[7] S. Leader,
Local proximity spaces , Math. Ann. (1967), 275–281.[8] S. MacLane,
Categories for the working mathematician , Springer-Verlag, New York, 1971, GraduateTexts in Mathematics, Vol. 5.[9] S. A. Naimpally and B. D. Warrack,
Proximity spaces , Cambridge Tracts in Mathematics and Mathe-matical Physics, No. 59, Cambridge University Press, London-New York, 1970.[10] J. Picado and A. Pultr,
Frames and locales: Topology without points , Frontiers in Mathematics,Birkh¨auser/Springer Basel AG, Basel, 2012., Frontiers in Mathematics,Birkh¨auser/Springer Basel AG, Basel, 2012.