In this paper, we first show that for any space X , there is a σ-complete Boolean subalgebra Z (Λ _X ) ^# of R ( X ) and that the subspace {α | α is a fixed σ Z ( X ) ^# -ultraiflter} of the Stone-space S ( Z (Λ _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. 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 (Λ _X ) ^# of R ( X ) and that the subspace {α | α is a fixed σ Z ( X ) ^# -ultraiflter} of the Stone-space S ( Z (Λ _X ) ^# ) is the minimal basically discon nected cover of X . Using this, we will show S ( Z (Λ _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 (Λ _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 ( X ) is basically disconnectedg, when partially ordered by inclusion, becomes a complete lattice. For the terminology, we refer to [1] and [9] . 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 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 ) = 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 ). 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 ) ^# = { cl_X ( int_X ( 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 s_B : S (𝓑) → βX , defined by s_B (α) = ∩{ cl_βX ( 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, int_X ( 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 f_U : 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 is a basically disconnected space, then ( Λ _βX ) is the minimal basically disconnected cover of X . For any covering map f : Y →X , let Z ( f ) ^# = { cl_Y ( int_X ( f ( Z ))) | Z ∈ Z ( Y ) ^# }. Since R (Λ X ) and R ( X ) are Boolean isomorphic and Z (Λ X ) ^# is a σ-complete Boolean subalgebra of R (Λ X ), by Lemma 2.1, Z (Λ _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} ≠ Let X be a space and for any Z (Λ _X ) ^# -ultrafilter α , let α _λ = { A ∈ Z (Λ X ) ^# | Λ _X ( A ) ∈ α }. Proposition 2.6 . Let X be a space and α a fixed Z (Λ _X ) ^# - ultrafilter. Then α_λ is a fixed Z(ΛX)#-ultrafilter and | ∩{A | A ∈ α _λ } |= 1. Proof. Clearly, α _λ is a Z (Λ X ) ^# -filter. Suppose that A ∈ Z (Λ X )# - α _λ . Then Λ _X ( A ) ∈ Z (Λ _X ) ^# - α . Since α is a Z (Λ X )#-ultrafilter, there is C ∈ α such that C ∧ Λ _X ( A ) = Ø and hence A ∧ cl _ΛX ( ( int_X ( C ))) = Since Λ _X ( cl_ΛX ( ( int_X ( C )))) = C ∈ α , cl _ΛX ( ( intX ( C ))) ∈ α ¸ and hence α _λ is a Z (Λ X ) ^# -ultrafilter. Since α is fixed, there is an x ∈ ∩{ B | B ∈ α }. Then { A ∩ ( x ) | A ∈ α _λ } has a family of closed sets in ( x ) with the finite intersection property. Since ( x ) is a compact subset of Λ X , ∩{A ∩ ( x ) | A ∈ α _λ } ≠ and hence ∩{A | A ∈ α _λ } ≠ . Since Z (Λ X ) ^# is a base for Λ X and α _λ ¸ is a Z (Λ X ) ^# -ultraifiter, | ∩{ A | A ∈ α _λ } |= 1. □ Let X be a space and FX = { α | α is a fixed Z (Λ _X ) ^# -ultrafilterg the subspace of the Stone space S ( Z (Λ _X ) ^# ). Define a map h_X : FX → Λ X by h_X ( α ) = ∩{A | A ∈ α _λ }. In the following, let Σ_B = for any B ∈ Z (Λ _X ) ^# . Theorem 2.7. Let X be a space. Then h_X : FX → Λ X is a homeomorphism. Proof. Take any α, δ in FX with α ≠δ. Since α and δ are Z (Λ _X ) ^# -ultrafilters, there are A , B in Z (Λ X ) ^# such that Λ _X ( A ) ∈ α , Λ _X ( B ) ∈ δ such that Λ _X ( A )∧Λ _X ( B ) = . Then A ∈ α _λ , B ∈ δ _λ ¸ and A ∧ B = . By Lemma 2.1, cl _ΛX ( A ) ∩ cl _ΛX ( B ) = and h_X ( α ) = ∩{ G | G ∈ α _λ } ≠ ∩{ H | H ∈ δ _λ } = hX ( δ ). Thus h_X is an one-to-one map. Let y ∈ Λ X and 𝛾 = {Λ _X ( C ) | y ∈ C ∈ Z (Λ X ) ^# }. Since every element of Z (Λ X ) ^# is a clopen set in Λ X , 𝛾 ∈ FX and h_X (𝛾) = y and hence h_X is an onto map. Let E ∈ Z (Λ X ) ^# . Suppose that µ ∈ FX - ( E ). Since Λ _X (E) ∉ µ, µ ∉ Σ _ΛX ( E ) and so Σ _ΛX ( E ) ⊆ h ^-1 (E). Suppose that θ ∈ ( E ). Then h_X ( θ ) ∈ E and hence for any A ∈ θ _λ A∩E ≠ . Since θ _λ is a Z (Λ X ) ^# -ultrafilter, E ∈ θ _λ and so E ∈Σ _ΛX(E) and h_X ( θ ) ∈ E . Hence Σ _ΛX ( E ) = ( E ). and since h_X is one-to-one and onto, h_X is a homeomorphism.          □ Corollary 2.8 . Let X be a space and F_X = ΛX ○ h_X. Then (FX, F_X) 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 (Λ _X ) ^# ) such that k ○ β_ΛX ○ h_X = j, where j : FX → S ( Z (Λ _X ) ^# ) is the inclusion map. Proof. By Theorem 2.7., βFX = βΛX and S ( Z (Λ _X ) ^# ) is a compactification of FX . Hence there is a continuous map k : βΛX → S ( Z (Λ _X ) ^# ) such that k○β_ΛX○h_X = j , where j : Λ X → S ( Z (Λ _X ) ^# ) is the dense embedding. Let T = S ( Z (Λ _X ) ^# ) and A, B be disjoint zero-sets in FX . Then there are disjoint zero-sets C, D in FX such that A ⊆ int_FX ( C ) and B ⊆ int _FX ( D ). Since h_X : FX → Λ _X is a homeomorphism, cl_FX ( int _FX ( C )) = Σ _FX ( cl_FX ( int_FX ( C ))) ∩ FX and since FX is dense in T , cl_T ( cl_FX ( int_FX ( C ))) = Σ _FX ( cl_FX ( int_FX ( C ))). Similarly, cl_T (cl_FX(int_FX(D))) = Σ_FX(cl_FX(in_tFX(D))). Since cl_FX ( int_FX ( C ))) ∧ cl_FX ( int_FX ( D ))) = , F_X(cl_FX(int_FX(C))) ∧ F_X(cl_FX(int_FX(D))) = . Hence cl_T(cl_FX(int_FX(C))) ∩ cl_T (cl_FX(int_FX(D))) = Σ_FX(cl_FX(int_FX(C))) ∩ Σ_FX(cl_FX(int_FX(D))) = Σ_FX(cl_ΛX(int_ΛX(C)))∧F_X(cl_FX(int_FX(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 (Λ 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 (Λ _X ) ^# = σZ ( X ) ^# . 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 ( A_n ) in M such that D = ∪{ | n ∈ N }. Clearly, cl_S ( M )( D ) ⊆ . Let α ∈ S ( M ) - cl_S ( _M )(∪{ | n ∈ N }). Then there is a B ∈ M such that α ∈ and (∪{ | n ∈ N ) ∩ = . Hence for any n ∈ N , ∩ = Σ _An∧B = . and hence A_n ∧ B = . So, ∨{ A_n ∧ B | n ∈ N } = (∨{ A_n | n ∈ N }) ∧ B = . Since B ∈ α , ∨{ A_n | n ∈ N } ∉ α and so α ∉ Σ _{∨{An|n∈N}} . Hence cl_S ( _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 Λ_β ○ t = s_M . 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 s_{Z (ΛX)#} ○ g = s_M if and only if Z (Λ _X ) ^# ⊆ M . (2) There is a covering map f : βΛX → S ( M ) such that s_M ○ f = s_{Z (ΛX)} ^# if and only if M ⊆ Z (Λ X ) ^# . (3) ( ( X ), s_MX ) is the minimal basically disconnected cover of X if and only if ( ( X ), s_MX ) is a basically disconnected cover of X and M ⊆ Z (Λ _X ) ^# . Proof. (1) (⇒) Take any Z ∈ Z (Λ 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 ) = . Since s_M and s_{Z (ΛX)#} are covering maps, cl_βX ( D ) = s_M ( g ^-1 ( A )) = s_{Z (ΛX)#} ( A ). By Lemma 2.1, D = s_M (g ^-1 ( A )) ∩ X = s_{Z (ΛX)#} ( 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 ( ( X ), s_MX ) is the minimal basically disconnected cover of X . Then there is a homeomorhpism l : ( X )→ Λ X such that Λ _X ○ l = sM_X . Hence there is a covering map f : βΛX → S ( M ) such that f ○ β_ΛX ○ 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 (Λ _X ) ^# such that Σ _A = f ^-1 ( ). Hence s_{Z (ΛX)^#} (Σ _A ) = cl_βX ( A ) = s_{Z (ΛX)^#} ( f ^-1 ( )). Since s_M ○ f ○ β_ΛX ○ ○ l = s_M ○ j = β_X ○ Λ_X ○ l = s_Z(ΛX)# ○ β_ΛX ○ l and β_ΛX ○ l is a dense embedding, s_M ○ f = s_{Z (ΛX)^#} . By (2), we have the result. (⇐)Since ( X ) is a basically disconnected space, there is a covering map l : ( X ) → Λ X such that Λ X ○ l = s_MX . Since M ⊆ Z (Λ _X ) ^# , by (2), there is a covering map f : βΛX → S ( M ) such that s_M ○ f = s_{Z (ΛX)^#} . Since s_M ○ f ○ β_ΛX = s_Z(ΛX)#○ β_ΛX = β_X○Λ_X , there is a covering m : Λ X → ( X ) such that s_MX ○m = Λ _X and j ○m = f ○ β_ΛX . Since Λ _X ○l ○m = s_MX ○m = Λ_X = Λ_X ○1Λ_X and Λ _X , l ○m are coevring maps, l ○ m = 1 _ΛX . Hence ( 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{ U_n | 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 𝒱 _n of 𝒰 _n such that G ⊆ cl_X (∪𝒱 _n ). 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 (Λ _X ) ^# is the top element.