In this paper, we first show that for any space
X
, there is a σcomplete Boolean subalgebra
Z
(Λ
_{X}
)
^{#}
of
R
(
X
) and that the subspace {α  α is a fixed σ
Z
(
X
)
^{#}
ultraiflter} of the Stonespace
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 σcomplete Boolean subalgebra of
R
(
X
) containing
Z(X)
# and
(
X
) is basically disconnectedg}, when partially ordered by inclusion, becomes a complete lattice.
1. INTRODUCTION
All spaces in this paper are Tychonoff spaces and
βX
denotes the StoneČech 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 Stonespace
S
(
σZ
(
X
)
^{#}
) of a σcomplete 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 Stonespace
S
(
σZ
(
X
)
^{#}
).
In this paper, we first show that for any space
X
, there is a σcomplete Boolean subalgebra
Z
(Λ
_{X}
)
^{#}
of
R
(
X
) and that the subspace {α  α is a fixed σ
Z
(
X
)
^{#}
ultraiflter} of the Stonespace
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 σcomplete Booeal subalgebra
M
of
R
(
X
) containing
Z
(
X
)
^{#}
, the Stonespace
S
(
M
) of
M
is a basically diconnected cover of X and that the subspace {α  α is a ﬁxed
M
ultraﬁlterg of the Stonespace
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 σcomplete 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
) =
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
σcomplete
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 σcomplete Boolean subalgebra
σM
of
L
containing
M
. Let
X
be a space and
Z
(
X
) the set of all zerosets 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 Stonespace of
𝓑. Then
S
(𝓑) is a compact, zerodimensional 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 zeroset
Z
in X,
int_{X}
(
Z
) is closed in
X
, equivalently, every cozeroset 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 σcomplete 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 Stonespace
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 corestriction 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 σcomplete Boolean subalgebra of
R
(Λ
X
), by Lemma 2.1,
Z
(Λ
_{X}
)
^{#}
is a σcomplete Boolean subalgebra of
R
(
X
).
Definition 2.5
. Let
X
be a space and 𝓑 a sublattice of
R
(
X
). Then a 𝓑filter 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 onetoone 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 onetoone 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 wellknown 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 zerosets in
FX
. Then there are disjoint zerosets
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
)
^{#}
.
3. BASICALLY DISCONNECTED COVERS
Let
X
be a space and
M
a σcomplete 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 σcomplete Boolean subalgebra of R
(
X
)
containg Z
(
X
)
^{#}
.
Then S
(
M
)
is a basically disconnected space.
Proof.
Let
D
be a
cozeroset
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
_{∨{Ann∈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 σcomplete 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 σcomplete 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 clopenset 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 σcomplete 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.
Adámek J.
,
Herrilich H.
,
Strecker G.E.
1990
Abstract and concrete categories
John Wiley and Sons Inc
New York
Gleason A.M.
1958
Projective topological spaces
Illinois J. Math.
2
Gillman L.
,
Jerison M.
1960
Rings of continuous functions
Van Nostrand
Princeton, New York
Henriksen M.
,
Vermeer J.
,
Woods R.G.
1987
QuasiFcovers of Tychonoff spaces
Trans. Amer. Math. Soc.
303
779 
804
Henriksen M.
,
Vermeer J.
,
Woods R.G.
1989
Wallman covers of compact spaces
Dissertationes Mathematicae
283
5 
31
Iliadis S.
1963
Absolute of Hausdorff spaces
Sov. Math. Dokl.
4
295 
298
Porter J.
,
Woods R.G.
1988
Extensions and Absolutes of Hausdorff Spaces
Springer
Berlin