FILTER SPACES AND BASICALLY DISCONNECTED COVERS

Abstract

In this paper, we first show that for any space X, there is a ${\sigma}$-complete Boolean subalgebra of $\mathcal{R}$(X) and that the subspace {${\alpha}{\mid}{\alpha}$ is a fixed ${\sigma}Z(X)^{\sharp}$-ultrafilter} of the Stone-space $S(Z({\Lambda}_X)^{\sharp})$ is the minimal basically disconnected cover of X. Using this, we will show that for any countably locally weakly Lindel$\ddot{o}$f space X, the set {$M{\mid}M$ is a ${\sigma}$-complete Boolean subalgebra of $\mathcal{R}$(X) containing $Z(X)^{\sharp}$ and $s_M^{-1}(X)$ is basically disconnected}, when partially ordered by inclusion, becomes a complete lattice.

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 (ΛX, ΛX) and if X is a compact space, then ΛX 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 σZ(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(ΛX)# of R(X) and that the subspace {α | α is a fixed σZ(X)#-ultraiflter} of the Stone-space S(Z(ΛX)#) is the minimal basically discon nected cover of X. Using this, we will show S(Z(ΛX)#) 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(ΛX)#. 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 (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 , 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 → , there is a continuous map g : Y → 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, = {α ∈ S(𝓑) | B ∈ α}. Then the space S(B), equipped with the topology for which { | 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 (ΛX, ΛX) and that if X is a compact space, then ΛX is the Stone-space S(σZ(X)#) of σZ(X)# and ΛX(α) =∩{A | A ∈ α} (α ∈ΛX).

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, (ΛβX, Λβ) denotes the minimal basically disconnected cover of βX.

Lemma 2.4 ([7]). Let X be a space. If is a basically disconnected space, then (ΛβX) 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(ΛX) and R(X) are Boolean isomorphic and Z(ΛX)# is a σ-complete Boolean subalgebra of R(ΛX), by Lemma 2.1, Z(ΛX)# 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} ≠

Let X be a space and for any Z(ΛX)#-ultrafilter α, let αλ = {A ∈ Z(ΛX)# | ΛX(A) ∈ α}.

Proposition 2.6. Let X be a space and α a fixed Z(ΛX)#-ultrafilter. Then αλ is a fixed Z(ΛX)#-ultrafilter and | ∩{A | A ∈αλ} |= 1.

Proof. Clearly, αλ is a Z(ΛX)#-filter. Suppose that A ∈ Z(ΛX)# - αλ. Then ΛX(A) ∈ Z(ΛX)# - α. Since α is a Z(ΛX)#-ultrafilter, there is C ∈ α such that C ∧ ΛX(A) = Ø and hence A∧clΛX( (intX(C))) = Since ΛX(clΛX((intX(C)))) = C ∈ α, clΛX((intX(C))) ∈ α¸ and hence αλ is a Z(ΛX)#-ultrafilter. Since α is fixed, there is an x ∈ ∩{B | B ∈ α}. Then {A∩ (x) | A ∈ αλ} has a family of closed sets in (x) with the finite intersection property. Since (x) is a compact subset of ΛX, ∩{A ∩ (x) | A ∈ αλ} ≠ and hence ∩{A | A ∈ αλ} ≠ . Since Z(ΛX)# is a base for ΛX and αλ¸ is a Z(ΛX)#-ultraifiter, | ∩{A | A ∈ αλ} |= 1. □

Let X be a space and FX = {α | α is a fixed Z(ΛX)#-ultrafilterg the subspace of the Stone space S(Z(ΛX)#). Define a map hX : FX → ΛX by hX(α) = ∩{A | A ∈ αλ}. In the following, let ΣB = for any B ∈ Z(ΛX)#.

Theorem 2.7. Let X be a space. Then hX : FX → ΛX is a homeomorphism.

Proof. Take any α, δ in FX with α ≠δ. Since α and δ are Z(ΛX)#-ultrafilters, there are A, B in Z(ΛX)# such that ΛX(A) ∈ α, ΛX(B) ∈ δ such that ΛX(A)∧ΛX(B) = . Then A ∈ αλ, B ∈ δλ¸ and A ∧ B = . By Lemma 2.1, clΛX(A) ∩ clΛX(B) = and hX(α) = ∩{G | G ∈ αλ} ≠ ∩{H | H ∈ δλ} = hX(δ). Thus hX is an one-to-one map.

Let y ∈ ΛX and 𝛾 = {ΛX(C) | y ∈ C ∈ Z(ΛX)#}. Since every element of Z(ΛX)# is a clopen set in ΛX, 𝛾 ∈ FX and hX(𝛾) = y and hence hX is an onto map.

Let E ∈ Z(ΛX)#. Suppose that µ ∈ FX-(E). Since ΛX(E) ∉ µ, µ ∉ΣΛX(E) and so ΣΛX(E) ⊆ h-1(E). Suppose that θ ∈ (E). Then hX(θ) ∈ E and hence for any A ∈ θλ A∩E ≠ . Since θλ is a Z(ΛX)#-ultrafilter, E ∈ θλ and so E ∈ΣΛX(E) and hX(θ) ∈ E. Hence ΣΛX(E) =(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(ΛX)#) such that k ○ βΛX ○ hX = j, where j : FX → S(Z(ΛX)#) is the inclusion map.

Proof. By Theorem 2.7., βFX = βΛX and S(Z(ΛX)#) is a compactification of FX. Hence there is a continuous map k : βΛX → S(Z(ΛX)#) such that k○βΛX○hX = j, where j : ΛX → S(Z(ΛX)#) is the dense embedding. Let T = S(Z(ΛX)#) 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 ⊆ intFX(D). Since hX : FX → ΛX is a homeomorphism, clFX(intFX(C)) = ΣFX(clFX(intFX(C))) ∩ FX and since FX is dense in T, clT (clFX(intFX(C))) = ΣFX(clFX(intFX(C))). Similarly, clT (clFX(intFX(D))) = ΣFX(clFX(intFX(D))).

Since clFX(intFX(C))) ∧ clFX(intFX(D))) = , 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 {ΛX(A) | A ∈ Z(ΛX)#} = σZ(X)#([5]). Hence we have the following :

Corollary 2.10. Let X be a space. Then βΛX = ΛβX if and only if Z(ΛX)# = 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 = ∪{ | n ∈N}. Clearly, clS(M)(D) ⊆ . Let α ∈ S(M) - clS(M)(∪{ | n ∈ N}). Then there is a B ∈ M such that α ∈ and (∪{ | n ∈ N) ∩ = . Hence for any n ∈ N, ∩ = ΣAn∧B = . and hence An ∧ B = . So, ∨{An ∧ B | n ∈ N} = (∨{An | n ∈ N}) ∧ B = . Since B ∈ α, ∨{An | n ∈ N} ∉ α and so α ∉ Σ∨{An|n∈N}. 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(ΛX)# ○ g = sM if and only if Z(ΛX)# ⊆ M.

(2) There is a covering map f : βΛX → S(M) such that sM ○ f = sZ(ΛX)# if and only if M ⊆ Z(ΛX)#.

(3) ((X), sMX) is the minimal basically disconnected cover of X if and only if ((X), sMX) is a basically disconnected cover of X and M ⊆ Z(ΛX)#.

Proof. (1) (⇒) Take any Z ∈ Z(ΛX)#. Then there is an A ∈ Z(βΛX)# such that Z = A ∩ ΛX. Since βΛX 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) = . 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(ΛX)#(A) ∩ X = ΛX(A ∩ ΛX) = ΛX(Z) and hence ΛX(Z) ∈ M.

(⇐) It is trivial([9]).

Similarly, we have (2)

(3) (⇒) Suppose that ( (X), sMX) is the minimal basically disconnected cover of X. Then there is a homeomorhpism l : (X)→ ΛX such that ΛX ○ l = sMX. Hence there is a covering map f : βΛX → S(M) such that f ○ βΛX ○ l = j, where j : (X) → S(M) is the inclusion map. Take any D ∈ M. Then f-1() is a clopen set in βΛX and since βΛX is a compact space, there is an A ∈ Z(ΛX)# such that ΣA = f-1(). Hence sZ(ΛX)#(ΣA) = clβX(A) = sZ(ΛX)#(f-1( )). Since sM ○ f ○ βΛX ○ ○ l = sM ○ j = βX ○ ΛX ○ l = sZ(ΛX)# ○ βΛX ○ l and βΛX ○ l is a dense embedding, sM ○ f = sZ(ΛX)#. By (2), we have the result.

(⇐)Since (X) is a basically disconnected space, there is a covering map l : (X) → ΛX such that ΛX ○ l = sMX. Since M ⊆ Z(ΛX)#, by (2), there is a covering map f : βΛX → S(M) such that sM ○ f = sZ(ΛX)#. Since sM ○ f ○ βΛX = sZ(ΛX)#○ βΛX = βX○ΛX, there is a covering m : ΛX → (X) such that sMX ○m = ΛX and j ○m = f ○ βΛX. Since ΛX ○l ○m = sMX ○m = ΛX = ΛX ○1ΛX and ΛX, l ○m are coevring maps, l ○ m = 1ΛX. Hence (X) and ΛX 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 ∈ } of open covers of X and any x ∈ X, there is a neighborhood G of x in X and for any n ∈ , there is a countable subset 𝒱n of 𝒰n such that G ⊆ clX(∪𝒱n).

It was shown that for any countably locally weakly Lindelöf space X, (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 (X) is basically disconnected}, when partially ordered by inclusion, becomes a complete lattice. Moreover, σZ(X)# is the bottom element and Z(ΛX)# is the top element.