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FILTER SPACES AND BASICALLY DISCONNECTED COVERS
FILTER SPACES AND BASICALLY DISCONNECTED COVERS
The Pure and Applied Mathematics. 2014. May, 21(2): 113-120
Copyright © 2014, Korean Society of Mathematical Education
  • Received : January 10, 2014
  • Accepted : April 10, 2014
  • Published : May 31, 2014
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About the Authors
YOUNG JU JEON
DEPARTMENT OF MATHEMATICS EDUCATION, CHONBUK NATIONAL UNIVERSITY, JEONJU 561- 756, REPUBLIC OF KOREAEmail address:jyj@jbnu.ac.kr
CHANGIL KIM
DEPARTMENT OF MATHEMATICS EDUCATION, DANKOOK UNIVERSITY, YONGIN 448-701, REPUB- LIC OF KOREAEmail address:kci206@hanmail.net

Abstract
In this paper, we first show that for any space X , there is a σ-complete Boolean subalgebra Z X ) # of R ( X ) and that the subspace {α | α is a fixed σ Z ( X ) # -ultraiflter} of the Stone-space S ( Z X ) # ) is the minimal basically disconnected cover of X. Using this, we will show that for any countably locally weakly Lindelöof space X , the set { M | M } is a σ-complete Boolean subalgebra of R ( X ) containing Z(X) # and ( X ) is basically disconnectedg}, when partially ordered by inclusion, becomes a complete lattice.
Keywords
1. INTRODUCTION
All spaces in this paper are Tychonoff spaces and βX denotes the Stone-Čech compactification of a space X .
Vermeer( [10] ) showed that every space X has the minimal basically disconnected cover (Λ X , Λ X ) and if X is a compact space, then Λ X is given by the Stone-space S ( σZ ( X ) # ) of a σ-complete Boolean subalgebra σZ ( X ) # of R ( X ). Henriksen, Vermeer and Woods( [4] )(Kim [7] , resp.) showed that the
minimal basically disconnected cover of a weakly Lindelöof space (a locally weakly Lindelöof space, resp.) X is given by the subspace {α | α is a fixed σ Z ( X ) # -ultrafilterg of the Stone-space S ( σZ ( X ) # ).
In this paper, we first show that for any space X , there is a σ-complete Boolean subalgebra Z X ) # of R ( X ) and that the subspace {α | α is a fixed σ Z ( X ) # -ultraiflter} of the Stone-space S ( Z X ) # ) is the minimal basically discon nected cover of X . Using this, we will show S ( Z X ) # ) and βΛX are homeomorphic. Moreover, we show that for any σ-complete Booeal subalgebra M of R ( X ) containing Z ( X ) # , the Stone-space S ( M ) of M is a basically diconnected cover of X and that the subspace {α | α is a fixed M -ultrafilterg of the Stone-space S ( M ) is the the minimal basically disconnected cover of X if and only if it is a basically disconnected space and M Z X ) # . Finally, we will show that for any countably locally weakly Lindelöof space X , the set { M | M } is a σ-complete Boolean subalgebra of R ( X ) containg Z ( X ) # and
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( X ) is basically disconnectedg, when partially ordered by inclusion, becomes a complete lattice.
For the terminology, we refer to [1] and [9] .
2. FILTER SPACES
The set R ( X ) of all regular closed sets in a space X , when partially ordered by inclusion, becomes a complete Boolean algebra, in which the join, meet, and complementation operations are defined as follows : for any A R ( X ) and any
  • {Ai:i∈I} ⊆R(X),
  • ∨{Ai:i∈I} =clX(∪{Ai : i ∈I}),
  • ∧{Ai:i∈I} =clX(intX(∩{Ai: i∈I})), and
  • A'=clX(X - A)
and a sublattice of R ( X ) is a subset of R ( X ) that contains
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, X and is closed under finite joins and meets.
We recall that a map f : Y X is called a covering map if it is a continuous, onto, perfect, and irreducible map.
Lemma 2.1 ( [5] ).
(1) Let f : Y→ X be a covering map. Then the map 𝜓: R ( Y ) → R ( X ), defined by 𝜓( A ) = f ( A ) ∩ X , is a Boolean isomorphism and the inverse map 𝜓 -1 of 𝜓 is given by 𝜓 -1 ( B ) = clY ( f -1 ( intX (B))) = clY ( intY ( f -1 ( B ))).
(2) Let X be a dense subspace of a space K. Then the map ϕ : R ( K ) → R ( X ), defined by ϕ(A) = A ∩ X, is a Boolean isomorphism and the inverse map ϕ-1 of ϕ is given by ϕ-1 ( B ) = clK ( B ).
A lattice L is called σ-complete if every countable subset of L has the join and the meet. For any subset M of a Boolean algebra L , there is the smallest σ-complete Boolean subalgebra σM of L containing M . Let X be a space and Z ( X ) the set of all zero-sets in X . Then Z ( X ) # = { clX ( intX ( Z )) | Z Z ( X )} is a sublattice of R ( X ).
We recall that a subspace X of a space Y is C*-embedded in Y if for any realvalued continuous map f : X
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, there is a continuous map g : Y
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such that g | x = f .
Let X be a space. Since X is C* -embedded in βX , by Lemma 2.1., σZ ( X ) # and σZ ( βX ) # are Boolean isomorphic.
Let X be a space and 𝓑 a Boolean subalgebra of R ( X ). Let S (𝓑) = {α | α is a B -ultrafilterg and for any B B ,
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= {α ∈ S (𝓑) | B ∈ α}. Then the space S ( B ), equipped with the topology for which {
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| B ∈ 𝓑} is a base, called the Stone-space of 𝓑. Then S (𝓑) is a compact, zero-dimensional space and the map sB : S (𝓑) → βX , defined by sB (α) = ∩{ clβX ( A ) | A ∈ 𝓑}, is a covering map ( [7] ).
Definition 2.2. A space X is called basically disconnected if for any zero-set Z in X, intX ( Z ) is closed in X , equivalently, every cozero-set in X is C* -embedded in X .
A space X is a basically disconnected space if and only if βX is a basically disconnected space. If X is a basically disconnected space, every element in Z ( X ) # is clopen in X and so X is a basically disconnected space if and only if Z ( X ) # is a σ-complete Boolean algebra.
Definition 2.3. Let X be a space. Then a pair ( Y, f ) is called
(1) a cover of X if f : Y → X is a covering map,
(2) a basically disconnected cover of X if ( Y, f ) is a cover of X and Y is a basically disconnected space, and
(3) a minimal basically disconnected cover of X if ( Y, f ) is a basically disconnected cover of X and for any basically disconnected cover ( Z, g ) of X , there is a covering map h : Z → Y such that f ○ h = g .
Vermeer( [10] ) showed that every space X has a minimal basically disconnected cover (Λ X , Λ X ) and that if X is a compact space, then Λ X is the Stone-space S ( σZ ( X ) # ) of σZ ( X ) # and Λ X ( α ) =∩{ A | A α } ( α ∈Λ X ).
Let X be a space. Since σZ ( X ) # and σZ ( βX ) # are Boolean isomorphic, S ( σZ ( X ) # ) and S ( σZ ( βX ) # ) are homeomorphic.
Let X, Y be spaces and f : Y → X a map. For any U X , let fU : f -1 ( U ) → U denote the restriction and co-restriction of f with respect to f -1 ( U ) and U , respectively.
In the following, for any space X , (Λ βX , Λ β ) denotes the minimal basically disconnected cover of βX .
Lemma 2.4 ( [7] ). Let X be a space. If
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is a basically disconnected space, then (
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Λ βX ) is the minimal basically disconnected cover of X .
For any covering map f : Y →X , let Z ( f ) # = { clY ( intX ( f ( Z ))) | Z Z ( Y ) # }. Since R X ) and R ( X ) are Boolean isomorphic and Z X ) # is a σ-complete Boolean subalgebra of R X ), by Lemma 2.1, Z X ) # is a σ-complete Boolean subalgebra of R ( X ).
Definition 2.5 . Let X be a space and 𝓑 a sublattice of R ( X ). Then a 𝓑-filter 𝓕 is called fixed if { F | ∈ 𝓕} ≠
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Let X be a space and for any Z X ) # -ultrafilter α , let α λ = { A Z X ) # | Λ X ( A ) ∈ α }.
Proposition 2.6 . Let X be a space and α a fixed Z X ) # - ultrafilter. Then αλ is a fixed Z(ΛX)#-ultrafilter and | ∩{A | A ∈ α λ } |= 1.
Proof. Clearly, α λ is a Z X ) # -filter. Suppose that A Z X )# - α λ . Then Λ X ( A ) ∈ Z X ) # - α . Since α is a Z X )#-ultrafilter, there is C ∈ α such that C ∧ Λ X ( A ) = Ø and hence A cl ΛX (
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( intX ( C ))) =
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Since Λ X ( clΛX (
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( intX ( C )))) = C α , cl ΛX (
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( intX ( C ))) ∈ α ¸ and hence α λ is a Z X ) # -ultrafilter. Since α is fixed, there is an x ∈ ∩{ B | B α }. Then { A
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( x ) | A α λ } has a family of closed sets in
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( x ) with the finite intersection property. Since
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( x ) is a compact subset of Λ X , ∩{A ∩
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( x ) | A ∈ α λ } ≠
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and hence ∩{A | A ∈ α λ } ≠
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. Since Z X ) # is a base for Λ X and α λ ¸ is a Z X ) # -ultraifiter, | ∩{ A | A α λ } |= 1. □
Let X be a space and FX = { α | α is a fixed Z X ) # -ultrafilterg the subspace of the Stone space S ( Z X ) # ). Define a map hX : FX → Λ X by hX ( α ) = ∩{A | A ∈ α λ }. In the following, let ΣB =
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for any B Z X ) # .
Theorem 2.7. Let X be a space. Then hX : FX → Λ X is a homeomorphism.
Proof. Take any α, δ in FX with α ≠δ. Since α and δ are Z X ) # -ultrafilters, there are A , B in Z X ) # such that Λ X ( A ) ∈ α , Λ X ( B ) ∈ δ such that Λ X ( A )∧Λ X ( B ) =
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. Then A ∈ α λ , B δ λ ¸ and A B =
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. By Lemma 2.1, cl ΛX ( A ) ∩ cl ΛX ( B ) =
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and hX ( α ) = ∩{ G | G α λ } ≠ ∩{ H | H δ λ } = hX ( δ ). Thus hX is an one-to-one map.
Let y ∈ Λ X and 𝛾 = {Λ X ( C ) | y C Z X ) # }. Since every element of Z X ) # is a clopen set in Λ X , 𝛾 ∈ FX and hX (𝛾) = y and hence hX is an onto map.
Let E Z X ) # . Suppose that µ FX -
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( E ). Since Λ X (E) ∉ µ, µ ∉ Σ ΛX ( E ) and so Σ ΛX ( E ) ⊆ h -1 (E). Suppose that θ
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( E ). Then hX ( θ ) ∈ E and hence for any A θ λ A∩E ≠
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. Since θ λ is a Z X ) # -ultrafilter, E θ λ and so E ∈Σ ΛX(E) and hX ( θ ) ∈ E . Hence Σ ΛX ( E ) =
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( E ). and since hX is one-to-one and onto, hX is a homeomorphism.          □
Corollary 2.8 . Let X be a space and FX = ΛX ○ hX. Then (FX, FX) is the minimal basically disconnected cover of X and F ( α ) = ∩{ A | A α } for all α FX .
It is well-known that a space X is C* -embedded in its compactification Y if and only if βX = Y .
Theorem 2.9. Let X be a space. Then there is a homeomorphism k : βΛX → S ( Z X ) # ) such that k ○ βΛX ○ hX = j, where j : FX → S ( Z X ) # ) is the inclusion map.
Proof. By Theorem 2.7., βFX = βΛX and S ( Z X ) # ) is a compactification of FX . Hence there is a continuous map k : βΛX S ( Z X ) # ) such that k○βΛX○hX = j , where j : Λ X S ( Z X ) # ) is the dense embedding. Let T = S ( Z X ) # ) and A, B be disjoint zero-sets in FX . Then there are disjoint zero-sets C, D in FX such that A intFX ( C ) and B int FX ( D ). Since hX : FX → Λ X is a homeomorphism, clFX ( int FX ( C )) = Σ FX ( clFX ( intFX ( C ))) ∩ FX and since FX is dense in T , clT ( clFX ( intFX ( C ))) = Σ FX ( clFX ( intFX ( C ))). Similarly, clT (clFX(intFX(D))) = ΣFX(clFX(intFX(D))).
Since clFX ( intFX ( C ))) ∧ clFX ( intFX ( D ))) =
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, FX(clFX(intFX(C))) ∧ FX(clFX(intFX(D))) = .
Hence clT(clFX(intFX(C))) ∩ clT (clFX(intFX(D))) = ΣFX(clFX(intFX(C))) ∩ ΣFX(clFX(intFX(D))) = ΣFX(clΛX(intΛX(C)))∧FX(clFX(intFX(D))) =
By the Uryshon’s extension theorem, FX is C* -embedded in T and so k is a homeomorphism.          □
It is known that βΛX = ΛβX if and only if {Λ X ( A ) | A Z X ) # } = σZ ( X ) # ( [5] ). Hence we have the following :
Corollary 2.10. Let X be a space. Then βΛX = ΛβX if and only if Z X ) # = σZ ( X ) # .
3. BASICALLY DISCONNECTED COVERS
Let X be a space and M a σ-complete Boolean subalgebra of R ( X ) containg Z ( X ) # . By the dfinition of σZ ( X ) # , σZ ( X )# ⊆ M .
Proposition 3.1. Let X be a space and M a σ-complete Boolean subalgebra of R ( X ) containg Z ( X ) # . Then S ( M ) is a basically disconnected space.
Proof. Let D be a cozero-set in S ( M ). Since S ( M ) is a compact space, D is a Lindelöf space and hence there is a sequense ( An ) in M such that D = ∪{
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| n N }. Clearly, clS ( M )( D ) ⊆
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. Let α S ( M ) - clS ( M )(∪{
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| n N }). Then there is a B M such that α
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and (∪{
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| n N ) ∩
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=
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. Hence for any n N ,
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= Σ An∧B =
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. and hence An ∧ B =
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. So, ∨{ An B | n N } = (∨{ An | n N }) ∧ B =
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. Since B α , ∨{ An | n N } ∉ α and so α ∉ Σ ∨{An|nN} . Hence clS ( M )( D ) is open in S ( M ) and thus S ( M ) is a basically disconnected space. □
Let X be a space and M a σ-complete Boolean subalgebra of R ( X ) containg Z ( X ) # . By Theorem 3.1, there is a covering map t : S ( M ) → ΛβX such that Λβ ○ t = sM .
Theorem 3.2. Let X be a space and M a σ-complete Boolean subalgebra of R ( X ) containg Z ( X ) # . Then we have the following :
(1) There is a covering map g : S ( M ) → βΛX such that sZ X)# g = sM if and only if Z X ) # M .
(2) There is a covering map f : βΛX → S ( M ) such that sM f = sZ X) # if and only if M Z X ) # .
(3) (
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( X ), sMX ) is the minimal basically disconnected cover of X if and only if (
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( X ), sMX ) is a basically disconnected cover of X and M ⊆ Z X ) # .
Proof. (1) (⇒) Take any Z Z X ) # . Then there is an A ∈ Z ( βΛX ) # such that Z = A ΛX . Since β Λ X is basically disconnected, g -1 ( A ) is a clopen-set in S (M ). Since S ( M ) is compact, there is a D M such that g -1 ( A ) =
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. Since sM and sZ (ΛX)# are covering maps, clβX ( D ) = sM ( g -1 ( A )) = sZ (ΛX)# ( A ). By Lemma 2.1, D = sM (g -1 ( A )) ∩ X = sZ X)# ( A ) ∩ X = Λ X ( A ∩ Λ X ) = Λ X ( Z ) and hence Λ X ( Z ) ∈ M .
(⇐) It is trivial( [9] ).
Similarly, we have (2)
(3) (⇒) Suppose that (
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( X ), sMX ) is the minimal basically disconnected cover of X . Then there is a homeomorhpism l :
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( X )→ Λ X such that Λ X l = sMX . Hence there is a covering map f : βΛX S ( M ) such that f ○ βΛX ○ l = j , where j :
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( X ) → S ( M ) is the inclusion map. Take any D M . Then f -1 (
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) is a clopen set in βΛX and since βΛX is a compact space, there is an A Z X ) # such that Σ A = f -1 (
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). Hence sZ X)# A ) = clβX ( A ) = sZ X)# ( f -1 (
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)). Since sM f ○ βΛX ○ ○ l = sM ○ j = βX ○ ΛX ○ l = sZ(ΛX)# ○ βΛX ○ l and βΛX ○ l is a dense embedding, sM ○ f = sZ X)# . By (2), we have the result.
(⇐)Since
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( X ) is a basically disconnected space, there is a covering map l :
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( X ) → Λ X such that Λ X l = sMX . Since M Z X ) # , by (2), there is a covering map f : βΛX → S ( M ) such that sM ○ f = sZ X)# . Since sM ○ f ○ βΛX = sZ(ΛX)#○ βΛX = βX○ΛX , there is a covering m : Λ X
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( X ) such that sMX ○m = Λ X and j ○m = f ○ βΛX . Since Λ X ○l ○m = sMX ○m = ΛX = ΛX ○1ΛX and Λ X , l ○m are coevring maps, l ○ m = 1 ΛX . Hence
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( X ) and Λ X are homeomorphic.          □
We recall that a space X is called a weakly Lindelöf space if for any open cover 𝒰, there is a countable subset 𝒱 of 𝒰 such that ∪𝒱 is dense in X and that X is called a countably locally weakly Lindelöf space if for any countable set{ Un | n
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} of open covers of X and any x X , there is a neighborhood G of x in X and for any n
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, there is a countable subset 𝒱 n of 𝒰 n such that G clX (∪𝒱 n ).
It was shown that for any countably locally weakly Lindelöf space X ,
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( X ) is a basically disconnected space( [8] ). Using Lemma 2.4 and Theorem 3.2, we have the following corollary :
Corollary 3.3 . Let X be a countably locally weakly Lindelöf space. Then the set { M | M is a σ-complete Boolean subalgebra of R ( X ) containg Z ( X ) # and
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( X ) is basically disconnected}, when partially ordered by inclusion, becomes a complete lattice. Moreover , σZ ( X ) # is the bottom element and Z X ) # is the top element.
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