In this paper, we investigate T1 Boolean subalgebras of the Boolean algebra consisting all regular closed sets in a space and their Wallman covers

http://dx.doi.org/10.5831/HMJ.2015.37.3.287

WALLMAN COVERS AND STONE- ˇCECH COMPACTIFICATIONS OF CLOZ COVERS

ChangIl Kim

Abstract. In this paper, we investigate T1 Boolean subalgebras of the Boolean algebra consisting all regular closed sets in a space and their Wallman covers. In particular, we will give sufficient and necessary conditions of spaces X satisfying βEcc(X) = Ecc(βX).

1. Introduction

All spaces in this paper are assumed to be Tychonoff spaces and (βX, β_X) denotes the Stone- ˇCech compactification of a space X.

Gleason [5] showed that the projective objects in the category of compact spaces and continuous maps are precisely the extremally dis- connected space and that every compact space has essentially unique projective cover, namely its extremally disconnected cover. Iliadis [9]

(Banaschewski [2], resp.) proved similar results for the category of Hausdorff spaces and perfect continuous maps (the category of regu- lar spaces and perfect continuous maps, resp.). To generalize extremally disconnected spaces, basically disconnected spaces, quasi-F spaces and cloz-spaces have been introduced and their minimal covers have been studied by various authors ¡

[3], [6], [7], [8], [10], [13]¢ .

Among others, Dashiell, Hager, and Henriksen ([3]) introduced the notion of cloz-spaces and they showed that every compact space X has a minimal cloz-cover (E_cc(X), z_X). In [10], it is shown that for a weakly Lindel¨of space X, βE_cc(X) = E_cc(βX) and every spaces has a minimal cloz-cover.

Further, in [11], authors investigated the relations with Wallman sub- lattices of the Boolean algebra R(X) of all regular closed sets in X con- tainig Z(X)^# = {cl_X(int_X(A)) | A is a zero-set in X} and quasi-F covers of a space X.

Received April 1, 2015. Accepted April 21, 2015.

2010 Mathematics Subject Classification. 54D80, 54G05, 54C10.

Key words and phrases. Wallman cover, cloz-space, covering map.

In this paper, we investigate T₁ Boolean subalgebras of the Boolean algebra consisting all regular closed sets in a space and their Wallman covers. In particular, we will give sufficient and necessary conditions for a space X satisfying βE_cc(X) = E_cc(βX).

For the terminology, we refer to [1], [4] and [12].

2. Wallman covers and Stone- ˇCech compactifications of cloz covers

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

{A_i | i ∈ I} ⊆ R(X),

∨{A_i | i ∈ I} = cl_X(∪{A_i | i ∈ I}),

∧{A_i | i ∈ I} = cl_X¡

int_X(∩{A_i | i ∈ I})¢ , and A⁰= cl_X(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. ([6])

(1) Let f : Y −→ X be a covering map. Then the map ψ : R(Y ) −→

R(X), defined by ψ(A) = A ∩ X, is a Boolean isomorphism and the inverse map ψ⁻¹ of ψ is given by ψ⁻¹(B) = cl_Y¡

f⁻¹(int_X(B))¢

= cl_Y¡

int_Y(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) = cl_K(B).

Throughout this paper, we assume that R is the space of all real numbers endowed with the usual topology.

We recall that a subset Z of a space X is called a zero-set in X if there is a continuous map f : Y −→ R such that Z = f⁻¹(0) and a subset of X is called a cozero-set in X if its complement is a zero-set in X.

Definition 2.2. Let X be a space.

(1) A cozero-set C in X is said to be a complemented cozero-set in X if there is a cozero-set D in X such that C ∩ D = ∅ and C ∪ D is a dense subset of X. In case, {C, D} is called a complemented pair of cozero-sets in X.

(2) Let G(X) = {cl_X(C) | C is a complemented cozero-set in X}.

A subspace Y of a space X is C^∗-embedded in X if for any continuous map f : Y −→ R, there is a continuous map g : X −→ R such that g|_Y = f .

Let X be a space and Z(X)^#= {cl_X(int_X(A)) | A is a zero-set in X}.

Suppose that {C, D} is a complemented pair of cozero-sets in X. Then cl_X(C) = cl_X(X −D) and since cl_X(X −D) ∈ Z(X)^#, cl_X(C) ∈ Z(X)^#. Hence G(X) = {A ∈ Z(X)^# | A⁰ ∈ Z(X)^#} and G(X) is a Boolean subalgebra of R(X).

Since X is C^∗-embedded in βX, by Lemma 2.1., G(X) and G(βX) are Boolean isomorphic.

Definition 2.3. ([7]) A space X is called a cloz-space if every element of G(X) is a clopen set in X.

We recall that a space X is a cloz-space if and only if βX is a cloz- space([7]).

Definition 2.4. Let X be a space.

(1) A pair (Y, f ) is called a cloz-cover of X if Y is a cloz-space and f : Y −→ X is a covering map.

(2) A cloz-cover (Y, f ) of X is called a minimal cloz-cover of X if for any cloz-cover (Z, g) of X, there is a covering map h : Z −→ Y with f ◦ h = g.

We recall that a sublattice A of R(X) is called

(a) T₁ if for any A ∈ A and x /∈ A, there is a B ∈ A such that x /∈ int_X(B) and A ∧ B = ∅,

(b) normal if for any A, B ∈ A with A ∧ B = ∅, there are C, D ∈ A such that A ∧ C = B ∧ D = ∅ and C ∪ D = X, and

(c) Wallman if A is T₁ normal.

Let X be a compact space, A a Wallman sublattice of R(X) and L(A) = {α | α is an A-ultrafilter}. Then {Σ^A_A | A ∈ A} is a base for closed sets of some compact topology on L(A). Let L(A, X) = {(α, x) | x ∈ α} be the subspace of L(A) × X and define a map ψ_A: L(A, X) −→

X by ψ_A((α, x)) = x. The pair (L(A, X), ψ_A) is called the Wallman cover of X with respect to A.

Then we have the following Lemma :

Lemma 2.5. [7] Let X be a compact space and A a Wallman sub- lattice of R(X). Then we have

(1) L(A, X) is a compact space,

(2) ψ_A : L(A, X) −→ X is a covering map, (3) for any A ∈ A, ψ_A

³

(Σ^A_A× X) ∩ L(A, X)

´

= A, and (4) for any A, B ∈ A with A ∧ B = ∅,

³

(Σ^A_A× X) ∩ L(A, X)

´

∩

³

(Σ^A_B× X) ∩ L(A, X)

´

= ∅

Let X be a compact space and B a Boolean subalgebra of R(X).

Then the space L(B) = S(B), equipped with the topology for which {Σ^B_B | B ∈ B} is a base, called the Stone-space of B. Moreover, {Σ^B_B | B ∈ B} is a base for closed sets in S(B) and hence S(B) is a compact zero-dimensional space([12]).

Henriksen, Vermeer and Woods showed that every compact space has the minimal cloz-cover (E_cc(X), z_X). Let X be a compact space. Then E_cc(X) = L(G(X), X) and z_X = ψ_G(X) ([7]). In ([10]), authors showed that every space X has a minimal cloz-cover (E_cc(X), z_X).

For any space X, let (E_cc(βX), z_β) denote the minimal cloz-cover of βX.

A space X is called a weakly Lindel¨of space if for any open cover U of X, there is a countable subfamily V of U such that ∪{V | V ∈ V} is a dense subset of X.

Let X be a weakly Lindel¨of space. Then E_cc(X) is the subspace {(α, x) ∈ L(G(X))×X | x ∈ ∩{A | A ∈ α}} of L(G(X))×X and E_cc(X) is a dense C^∗-embedded subspace of E_cc(βX), i.e., βE_cc(X) = E_cc(βX) ([10]).

For any space X, let G(z_X) = {cl_Y(int_X(z_X(A))) | A ∈ G(E_cc(X))}.

Since R(E_cc(X)) and R(X) are isomorphic and G(E_cc(X)) is a Boolean algebra, by Lemma 2.1, G(z_X) is a Boolean subalgebra of R(X). Since z_X : E_cc(X) −→ X is covering map, G(X) ⊆ G(z_X).

Let B be a Boolean subalgebra of R(X) with G(X) ⊆ B and let B_β = {cl_βX(B) | B ∈ B}, K(B_β) = {(α, x) | x ∈ ∩α}, ψ_Bβ = l_B. For any map f : Y −→ X and U ⊆ X, let f_U : f⁻¹(U ) −→ U be the restriction and corestriction of f with respect to U and f⁻¹(U ), respectively.

Lemma 2.6. Let X be a space and B a Boolean subalgebra of R(X) with G(X) ⊆ B. Then we have the following :

(1) ¡

K(B_β), l_B¢

is a cover of βX and there is a covering map f_B : K(B_β) −→ E_cc(βX) such that l_B = z_β ◦ f_B,

(2) if (l⁻¹_B (X), l_BX) is the minimal cloz cover of X, then B ⊆ G(z_X), and

(3) if (l_B⁻¹(X), l_BX) is a cloz cover of X and B ⊆ G(z_X), then (l_B⁻¹(X), l_BX) is the minimal cloz cover of X.

Proof. (1) Since G(βX) is a T₁ sublattice of R(βX) and G(X)_β = G(βX) ⊆ B_β, B_β is a T₁ sublattice of R(βX). Since B is a Boolean subalgebra of R(X), B_β is normal and so B_β is a Wallman sublatice of R(βX). By Lemma 2.5, ¡

K(B_β), l_B¢

is a cover of βX and since G(βX) ⊆ B_β, there is a covering map f_B : K(B_β) −→ E_cc(βX) such that l_B = z_β◦ f_B.

(2) Suppose that (l⁻¹_B (X), l_BX) is the minimal cloz cover of X. Then there is a homeomorphism g : l_B⁻¹(X) −→ E_cc(X) such that z_X◦g = l_BX.

Now, we claim that B ⊆ G(z_X). Take any B ∈ B. Let H = Σ^B_cl^β

βX(B). Since cl_βX(B) ∈ B_β and B is a Boolean algera,

cl_βX(B)⁰= cl_βX

³

βX − cl_βX(B)

´

∈ B_β and by Lemma 2.5,

³

H × βX

´

∩ K(B_β) is a clopen set in K(B_β). Hence g

³

(H × βX) ∩ l_B⁻¹(X)

´

∈ G(E_cc(X)) and z_X

³

g((H × βX) ∩ l⁻¹_B (X))

´

= l_BX

³

(H × βX) ∩ l_B⁻¹(X)

´

= l_B

³

(H × βX)

´

∩ X

= cl_βX(B) ∩ X = B.

Thus B ∈ G(z_X) and so B ⊆ G(z_X).

(3) Suppose that (l⁻¹_B (X), l_BX) is a cloz cover of X and B ⊆ G(z_X).

Note that there is a continuous map k : βE_cc(X) −→ E_cc(βX) such that z_β ◦ k ◦ β_Ecc(X) = β_X ◦ z_X. By (1), there is a continuous map f_B : K(B_β) −→ E_cc(βX) such that z_β◦ f_B = l_B. Since B_β ⊆ G(z_X)_β, there is a continuous map h : βE_cc(X) −→ K(B_β) such that l_B◦h = z_β◦k.

Let j₁: l_B⁻¹(X) −→ K(B_β) be the inclusion map. Then l_B◦ h ◦ β_Ecc(X) =

β_X ◦ z_X, there is a continuous map m : E_cc(X) −→ l_B⁻¹(X) such that l_BX◦ m = z_X and j₁◦ m = h ◦ β_Ecc(X).

We claim that m is a covering map. Let (α, x) ∈ l⁻¹_B (X). Since h is an onto map, there is a p ∈ βE_cc(X) such that h(p) = (α, x) and since l_BX is a covering map and h(p) = (α, x) ∈ l⁻¹_β (X), l_B(h(p)) ∈ X.

Hence l_B(h(p)) = l_BX(h(p)) = z_X(p) and since z_X is a covering map, p ∈ E_cc(X). Thus m(p) = h(p) = (α, x) and m is onto. Since l_BX ◦ m = z_X is a covering map and m is onto, m is a covering map([12]).

Since (l⁻¹_B (X), l_BX) is a cloz cover of X, there is a covering map d : l_B⁻¹(X) −→ E_cc(X) such that z_X◦ d = l_BX and so

z_X◦ d ◦ m = l_BX◦ m = z_X = z_X ◦ 1_Ecc(X).

Since z_X and d ◦ m are covering map, d ◦ m = 1_Ecc(X) and hence m : E_cc(X) −→ l⁻¹_B (X) is a homeomorphism. Thus one has the result

Let X be a space and B a Boolean subalgebra of R(X) such that G(X) ⊆ B and G(X) is a base for closed sets in X. Let

s(X, B) = {α | α is fixed a B −ultrafilter}

which is the subspace of the Stone space S(B) of B and for any G(z_X)- ultrafilter α,

α_c= {A ∈ G(βX) | A ∩ X ∈ α}.

Lemma 2.7. Let X be a space such that G(X) is a base for closed sets in X. Then the map s_X : s(X, G(z_X)) −→ z_β⁻¹(X), defined by s_X(α) = (α_c, ∩α_c), is a homeomorphism.

Proof. Let α ∈ s(X, B). Since G(X) = G(βX)∩X and G(X) ⊆ G(z_X), α_c is a G(βX)-ultrafilter. Since G(X) is a base for closed sets in X, B is a base for closed sets in X. Hence | ∩α |= 1 and since ∩α ∈ X,

| ∩α_c|= 1. Thus s_X : s(X, G(z_X)) −→ z_β⁻¹(X) is well-defined.

Clearly, s_X is an one-to-one map. Now, we claim that s_X is onto.

Let (α, t) ∈ z⁻¹_β (X) and

γ = {C ∈ G(z_X) | cl_βX(C) ∈ α}.

Then γ is a G(z_X)-filter. Let D ∈ G(E_cc(X)) such that z_X(D) /∈ γ.

Then cl_βX(z_X(D)) /∈ α and since α is a G(βX)-ultrafilter, there is an E ∈ G(βX) such that E ∈ α and cl_βX(z_X(D)) ∧ E = ∅. Hence z_X(D) ∧ (E ∩ X) = ∅ and E ∩ X ∈ G(X). Since G(X) ⊆ G(z_X), E ∩ X ∈ G(z_X) and since cl_βX(E ∩ X) = E ∈ α, by the definition of γ, E ∩ X ∈ γ.

Thus γ is a G(z_X)-ultrafilter. Since G(X) is a base for closed sets in X, {t} = ∩α and γ_c= α. Hence s_X(γ) = (α, t) and so s_X is onto.

Let µ ∈ s(X, G(z_X)) and s_X(µ) = (µ_c, ∩µ_c) ∈

³

Σ^G(z_cl ^X)β

βX(A) × βX

´

∩ z_β⁻¹(X) for some A ∈ G(βX). Note that

³

Σ^G(z_cl ^X)β

βX(A)× βX

´

∩ z_β⁻¹(X) is a clopen set in z_β⁻¹(X) and

s_X

³

Σ^G(βX)_A ∩ s(X, B)

´

⊆

³

Σ^G(z_cl ^X)β

βX(A)× βX

´

∩ z_β⁻¹(X).

Hence s_X is continuous.

Finally, we will show that s⁻¹_X is continuous. Let (θ, t) ∈ z_β⁻¹(X) and s⁻¹_X ((θ, t)) ∈ Σ^G(βX)_cl

βX(F )∩ s(X, B) for some F ∈ G(z_X). Then cl_βX(F ) ∈ θ and since

s⁻¹_X

³³

Σ^G(z_cl ^X)β

βX(F )× βX

´

∩ z⁻¹_β (X)

´

⊆ Σ^G(βX)_cl

βX(F )∩ s(X, B), s⁻¹_X is continuous.

Using Lemma 2.7, we have the following corollary : Corollary 2.8.

(1) Let X be a space and B a Boolean subalgebra of R(X) such that G(X) ⊆ B and G(X) is a base for closed sets in X. Then s(X, B) and l_B⁻¹(X) are homeomorphic.

(2) Let X be a locally compact zero-dimensional space. Then s(X, G(z_X)) and E_cc(X) are homeomorphic and l⁻¹_G(z

X)(X) = s(X, G(z_X)).

(3) Let X be a locally compact zero-dimensional space. Then z_β⁻¹(X) and E_cc(X) are homeomorphic.

Proof. (1) Similar to the proof of Lemma 2.7, one has the result. (2) Let A, B ∈ G(z⁻¹_β (X)) with A ∧ B = ∅. Suppose that A ∩ B 6= ∅.

Pick p ∈ A ∩ B. Then z_β(p) ∈ X and since X is locally compact zero- demensional, there is a clopen set V in X such that V is a compact subset of X and z_β(p) ∈ V . Hence V is clopen βX and since z_β is continuous, z_β⁻¹(V ) is a clopen set in E_cc(βX) and p ∈ z_β⁻¹(V ). Since E_cc(βX) is a cloz-space, z_β⁻¹(V ) is a cloz-space. Let W = z_β⁻¹(V ). Then A ∩ W, B ∩ W ∈ G(W ) and (A ∩ W ) ∧ (B ∩ W ) = ∅. Since W is a colz space, (A ∩ W ) ∩ (B ∩ W ) = ∅ whch is a contradiction to p ∈ A ∩ B ∩ W . Thus A ∩ B = ∅ and so z⁻¹_β (X) is a cloz space. Since G(X) ⊆ G(z_X), by (3) in Lemma 2.6, (z_β⁻¹(X), z_βX) is the minimal cloz cover of X.

By Lemma 2.7 and (1) in this corollary, l⁻¹_G(z

X)(X) = s(X, G(z_X)).

(3) Silmilar to the proof in (2), one has (3).

For any space X, let K(X) = {B | B is a Boolean subalgebra of R(X) such that G(X) ⊆ B and (l⁻¹_B (X), l_BX) is the minimal cloz cover of X}.

Using Lemma 2.6 and Corollary 2.8, we get the following theorm.

Theorem 2.9. Let X be a locally compact zero-dimensional space.

Then (K(X), ⊆) is a complete lattice with the bottom element G(X) and the top element G(z_X).

Proof. By Corollary 2.8, G(X) is the bottom element in (K(X), ⊆) and G(z_X) is the top element in (K(X), ⊆) . Take any B and C in K(X).

Then there is the smallest Boolean subalgebra E of R(X) containing B and C and G(X) ⊆ E ⊆ G(z_X). Hence G(βX) ⊆ E_β ⊆ G(z_X)_β and thus there are covering map

g₁: K(G(z_X)_β) → K(E_β), g₂: K(E_β) → E_cc(βX)

such that l_G(zX)= l_E ◦ g₁ and l_E = z_β◦ g₂. Since G(X), G(z_X) ∈ K(X), l⁻¹_G(z

X)(X) = E_cc(X) = z_β⁻¹(X) and so

l⁻¹_E (X) = E_cc(X).

Hence E ∈ K(X) and E = B ∨ C in (K(X), ⊆). Similarly B ∧ C = B ∩ C in (K(X), ⊆). Thus (K(X), ⊆) is a lattice. Similarly, we can show that (K(X), ⊆) is a complete upper semi-lattice and that (K(X), ⊆) is a complete lattice(see ([12] for the details).

It is well-known that βX is a zero-dimensional space if and only if for any disjoint zero-sets A, B in X, there is a clopen set C in X such that A ⊆ C and B ∩ C = ∅([12]).

Theorem 2.10. Let X be a locally compact zero-dimensional space.

Then the following are equivalent : (1) K(X) = {G(X)},

(2) βE_cc(X) = E_cc(βX), and

(3) βE_cc(X) is a zero-dimensional space.

Proof. (1) ⇒ (2) Let C = G(βE_cc(X)). Then C is a Boolean subalge- bra of R(βE_cc(X)) such that

G(E_cc(X)) = C ∩ E_cc(X)

Let k : βE_cc(X) −→ E_cc(βX) be the covering map such that z_β◦ k ◦ β_Ecc(X)= z_X◦ β_X. Consider the cover

³

L(C, βE_cc(X)), ψ_C

´

of E_cc(βX).

Since βE_cc(X) is a cloz space, the map ψ_C : L(C, βE_cc(X)) −→ βE_cc(X) is a homeomorphism and ψ_C(α, p) = p.

Now, we claim that k ◦ ψ_C: L(C, βE_cc(X)) −→ E_cc(βX) is an one-to- one map. Take any (α, p), (δ, q) in L(C, βE_cc(X)) with (α, p) 6= (δ, q).

Suppose that p 6= q. Since ψ_C is an one-to-one map, α 6= δ. Hence (α, p) 6= (δ, q) implies α 6= δ. Since α and δ are C-ultrafilters, there are A, B ∈ C such that A ∈ α, B ∈ δ and A ∧ B = ∅. Note that

(z_β◦ k ◦ ψ_C)(C) = (z_β ◦ k ◦ ψ_C)(G(E_cc(X))_β) = G(z_X)_β. By Theorem 2.9 and (1) in this theorem,

(z_β ◦ k ◦ ψ_C)(C) = G(βX) and so

z_β(k(ψ_C(A))), z_β(k(ψ_C(A))) ∈ G(βX).

By Lemma 2.5,

k ◦ ψ_C(A) ∩ k ◦ ψ_C(B) = ∅.

Since k ◦ ψ_C(α, p) ∈ k ◦ ψ_C(A) and k ◦ ψ_C(δ, q) ∈ k ◦ ψ_C(B), k ◦ ψ_C(α, p) 6=

k ◦ ψ_C(δ, q) and hence k ◦ ψ_C : L(C, βX) −→ E_cc(βX) is an one-to-one map.

(2) ⇒ (3) Let G be an open set in E_cc(βX) and (α, x) ∈ G. Then there is an A ∈ G(βX) such that (α, x) ∈ (Σ^G(βX)_A ×βX)∩E_cc(βX) ⊆ G.

Since (Σ^G(βX)_A × βX) ∩ E_cc(βX) is a clopen set in E_cc(βX), E_cc(βX) is a zero-dimensional space and thus βE_cc(X) is a zero-dimensional space.

(3) ⇒ (2) Since βE_cc(X) is a zero-dimensional space, G(βE_cc(X)) is a base for closed sets in βE_cc(X) and G(βE_cc(X)) is a base for βE_cc(X).

We claim that E_cc(X) is C^∗-embedded in E_cc(βX). Take any disjoint zero-sets Z₁, Z₂ in E_cc(X). Then there are disjoint zero-sets A, B in E_cc(X) such that Z₁ ⊆ int_Ecc(X)(A) and Z₂ ⊆ int_Ecc(X)(B) . Since βE_cc(X) is a zero-dimensional space, there is an clopen C ∈ E_cc(X) such that

A ⊆ C, B ∩ C = ∅.

Since C is clopen in E_cc(X), C ∈ G(E_cc(X)) and so cl_Ecc(βX)(C) is clopen in E_cc(βX). Note that

int_Ecc(βX)(B) ∩ cl_Ecc(βX)(C) ⊆ cl_Ecc(X)

³

int_Ecc(βX)(B) ∩ cl_Ecc(βX)(C)

´

= cl_Ecc(X)

³

int_Ecc(βX)(B) ∩ C

´

= ∅.

Since cl_Ecc(βX)(C) is clopen in E_cc(βX), we have cl_Ecc(βX)

³

int_Ecc(βX)(B)

´

∩ cl_Ecc(βX)(C) = ∅,

and so cl_Ecc(βX)(Z₁) ∩ cl_Ecc(βXZ₂) = ∅. Hence E_cc(βX) is C^∗-embedded in E_cc(βX) and thus one have (2).

(2) ⇒ (1) Take any B in K(X). Then G(X) ⊆ B and (l⁻¹_B (X), l_BX) is the minimal cloz cover of X, that is, l_B⁻¹(X) = E_cc(X). By Lemma 2.6, B ⊆ G(z_X). Similar to the proof in (3) ⇒ (2), we can show that E_cc(X) is C^∗-embedded in L(G(z_X)_β, βX) and so L(G(z_X)_β, βX) = βE_cc(X) = E_cc(βX). Hence G(z_X)_β = G(βX) and thus G(X) = G(z_X). Therefore, one has (1).

It is known that for any weakly Lindel¨of space X, βE_cc(X) = E_cc(βX) and by Theorem 2.10, we have the following corollary :

Corollary 2.11. Let X be a locally compact zero-dimensional, weakly Lindel¨of space. Then K(X) = {G(X)} and βE_cc(X) is a zero-dimensional space.

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ChangIl Kim

Department of Mathematics Education, Dankook University, 126, Jukjeon, Yongin, Gyeonggi, South Korea 448-701 , KOREA.

E-mail: kci206@hanmail.net