Download Mohammad S. Sarsak B-CLOSED SPACES 1. Introduction and

DEMONSTRATIO MATHEMATICA
Vol. XLV
No 1
2012
Mohammad S. Sarsak
B-CLOSED SPACES
Abstract. We introduce and study B-closed spaces. This class of spaces is a subclass
of both S-closed spaces and p-closed spaces.
1. Introduction and preliminaries
A subset A of a space X is called regular open if A = Int A, and regular
closed if X\A is regular open, or equivalently, if A = Int A; it is well known
that Int A (respectively, Int A) is a regular open (resp. regular closed) subset
of X for every subset A of a space X. A is called semi-open [15] (resp.
preopen [16], b-open [4], α-open [18]) if A ⊂ Int A (resp. A ⊂ Int A, A ⊂
Int A ∪ Int A, A ⊂ Int Int A). A is called semi-closed (resp. preclosed, bclosed, α-closed) if X\A is semi-open (resp. preopen, b-open, α-open), or
equivalently, if Int A ⊂ A (resp. Int A ⊂ A, Int A ∩ Int A ⊂ A, Int A ⊂
A). Ganster [11] pointed out that a set is preopen if and only if it is the
intersection of an open set with a dense set. It is also known that any
union of semi-open (resp. preopen, b-open) sets is semi-open (resp. preopen,
b-open). The preclosure (resp. b-closure) of A denoted by pcl A (resp. bcl)
is the intersection of all preclosed (resp. b-closed) subsets of X containing
A, or equivalently, the smallest preclosed (resp. b-closed) set containing A.
The preinterior of A denoted by pint A is the union of all preopen subsets of
X contained in A, or equivalently, the largest preopen set contained in A.
Andrijević [3] pointed out that pcl A = A ∪ Int A, he observed in [4] that
A is preopen if and only if A ⊂ pint (pcl A) and that A is b-open if and only if
A ⊂ pcl (pint A), thus, it is easy to see that A is preclosed (resp. b-closed) if
and only if pcl (pint A) ⊂ A (resp. pint(pcl A) ⊂ A). A is called pre-regular
p-open (resp. pre-regular p-closed) [13] if A = pint (pcl A) (resp. X\A is
pre-regular p-open, or equivalently, if A = pcl (pint A)). It is easy to see
2000 Mathematics Subject Classification: Primary 54D20.
Key words and phrases: regular open, semi-open, preopen, α-open, b-open, pre-regular
p-open, regular p-open, S-closed space, p-closed space, B-closed space, B-closed set.
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that pint (pcl A) (resp. pcl (pint A)) is pre-regular p-open (resp. pre-regular
p-closed) for any subset A of a space X. A is called regular p-open [13]
if there exists a pre-regular p-open set U with U ⊂ A ⊂ pcl U . A is called
predense [13] if pcl A = X, it was shown in [21] that A is predense if and only
if Int A is dense, thus predense sets are α-open. We will denote the families of
regular open (resp. semi-open, preopen, b-open, α-open, pre-regular p-open,
regular p-open) subsets of a space X by RO (X) (resp. SO (X), P O (X),
BO (X), αO (X), P RP O (X) , RP O (X)). We will also denote the families
of regular closed (resp. semi-closed, preclosed, b-closed, α-closed, pre-regular
p-closed) subsets of a space X by RC (X) (resp. SC (X), P C (X), BC (X),
αC (X), P RP C (X)). Njastad [18] pointed out that the family of all α-open
subsets of a space (X, τ ), denoted by τ α , is a topology on X finer than τ .
Reilly and Vamanamurthy observed in [20] that τ α = SO (X) ∩ P O (X).
If (X, τ ) is a topological space, we will denote the space (X, τ α ) by X α .
Jankovic [14] pointed out that P O (X) = P O (X α ), SO (X) = SO (X α ),
α
and αO (X) = αO (X α ). Andrijević observed in [2] that Int A = Intα A
α
and Int A = Intα A for any subset A of a space X, he observed in [4]
that SO (X) ∪ P O (X) ⊂ BO (X) and that BO (X) = BO (X α ). Sarsak
observed in [21] that RO (X) ⊂ P RP O (X) ⊂ RP O (X), RP O (X) =
BO (X) ∩ BC (X), P RP O (X) = P O (X) ∩ BC (X) and that P RP O (X)
= P RP O (X α ) and RP O (X) =RP O (X α ). It is also known that the intersection of a semi-open (resp. preopen, b-open) set and an α-open set is
semi-open (resp. preopen, b-open).
A space X is called strongly compact [17] (resp. semi-compact [7]) if any
preopen (resp. semi-open) cover of X has a finite subcover. Ganster [12]
pointed out that a space that is both strongly compact and semi-compact
is finite. X is called p-closed [1] if any preopen cover of X has a finite
subfamily, the union of the preclosures of whose members covers X. X is
called S-closed [23] if any semi-open cover of X has a finite subfamily, the
union of the closures of whose members covers X, or equivalently [5], if every
regular closed cover of X has a finite subcover. A subset A of a space X is
called strongly compact in X [17] if any cover of A by preopen subsets of X
has a finite subcover of A. A is called p-closed in X [8] if any cover of A by
preopen subsets of X has a finite subfamily, the union of the preclosures (in
X) of whose members covers A, and A is called S-closed in X if any cover of
A by semi-open subsets of X has a finite subfamily, the union of the closures
(in X) of whose members covers A.
A space X is called resolvable if there is a subset D of X such that D and
X\D are both dense in X. A subset A of X is resolvable if it is resolvable as
a subspace. A space X is called irresolvable if it is not resolvable. A space X
is said to be strongly irresolvable [10] if every nonempty open subset of X is
B-closed spaces
205
irresolvable, or equivalently [11], if every preopen subset of X is semi-open.
A space X is called extremally disconnected [22] if the closure of each open
subset of X is open, or equivalently [14], if every semi-open subset of X is
preopen.
A space X is called locally indiscrete if every open subset of X is closed.
A function f from a space X into a space Y is called irresolute [6] if the
inverse image of each semi-open subset of Y is semi-open in X, and called
preirresolute [20] if f −1 (V ) is preopen in X for every preopen subset V of
Y , or equivalently [19], if f (pcl A) ⊂ pcl f (A) for every subset A of X.
Throughout this paper, a space X stands for a topological space. If
A ⊂ X, where X is a space, then A and Int A denote respectively, the closure
α
of A in X and the interior of A in X, A and Intα A denote respectively, the
closure of A in X α and the interior of A in X α . If A ⊂ B ⊂ X, where X
B
is a space, then A , pclB A and IntB A denote respectively, the closure of A
in B, the preclosure of A in B and the interior of A in B. For the concepts
not defined here, we refer the reader to [9].
In concluding this section, we recall the following facts from [21] for their
importance in the material of our paper.
Proposition 1.1. A subset A of a space X is b-open if and only if A =
B ∪ C, where B is semi-open and C is preopen.
Proposition 1.2.
(i) Let A and B be subsets of a space X such that A ⊂ B. If A ∈ BO (X),
then A ∈ BO (B).
(ii) If A ∈ BO (B), B ∈ αO (X), then A ∈ BO (X).
Proposition 1.3. A space X is extremally disconnected if and only if
every b-open subset of X is preopen.
Proposition 1.4. A space X is strongly irresolvable if and only if every
b-open subset of X is semi-open.
Proposition 1.5. For a space X, the following are equivalent:
(i) X is locally indiscrete,
(ii) Every b-open subset of X is preclosed.
2. B-closed spaces
Definition 2.1. A space X is called B-closed if any b-open cover of X has
a finite subfamily, the union of the preclosures of whose members covers X.
Remark 2.2. Since SO (X) ∪ P O (X) ⊂ BO (X), and since pcl A = A
whenever A is semi-open, it is clear that every B-closed space is both S-
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closed and p-closed. However, the author asks about the existence of a space
that is both S-closed and p-closed but not B-closed.
The following two propositions follow from Propositions 1.3 and 1.4 and
from the fact that pcl A = A whenever A is semi-open.
Proposition 2.3. For an extremally disconnected space X, the following
are equivalent:
(i) X is B-closed,
(ii) X is p-closed.
Proposition 2.4. For a strongly irresolvable space X, the following are
equivalent:
(i) X is B-closed,
(ii) X is S-closed.
The following result is an immediate consequence of Proposition 1.1 and
from the fact that SO (X) ∪ P O (X) ⊂ BO (X).
Proposition 2.5. A space X is B-closed if and only if any cover of X
whose members are semi-open or preopen has a finite subfamily, the union
of the preclosures of whose members covers X.
Lemma 2.6. A subset A of a space X is b-open if and only if there exists
a preopen subset U of X such that U ⊂ A ⊂ pcl U .
Proposition 2.7. For a space X, the following are equivalent:
(i) X is B-closed,
(ii) any regular p-open cover of X has a finite subfamily, the union of the
preclosures of whose members covers X,
(iii) any pre-regular p-closed cover of X has a finite subcover.
Proof. (i)→(ii): Follows since every regular p-open set is b-open.
(ii)→(iii): Follows since every pre-regular p-closed set is regular p-open
and preclosed.
(iii)→(i): Let U = {Uα : α ∈ Λ} be a b-open cover of X. Then by
Lemma 2.6, for each α ∈ Λ, there exists a preopen subset Vα of X such
that Vα ⊂ Uα ⊂ pcl Vα . Now V = {pcl Vα : α ∈ Λ} is pre-regular p-closed
coverSof X and thus
Snby (ii), there exists α1 , α2 , . . . , αn ∈ Λ such that
n
X = i=1 pcl Vαi = i=1 pcl Uαi . Hence, X is B-closed.
The following result follows from the the definition of a B-closed space
and from Propositions 2.5 and 2.7, the straightforward proof is omitted.
Proposition 2.8. For a space X, the following are equivalent:
(i) X is B-closed,
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(ii) for
T any family U = {Uα : α ∈ Λ} of b-closed subsets of X such that
T {Uα : α ∈ Λ} = ∅, there exists a finite subset Λ0 of Λ such that
{pint Uα : α ∈ Λ0 } = ∅,
(iii) for any family U = {Uα : α ∈ T
Λ} each of whose members is semi-closed
or preclosed in X such that
{Uα : α ∈ Λ} = ∅, there exists a finite
T
subset Λ0 of Λ such that {pint Uα : α ∈ Λ0 } = ∅,
(iv) for any
T family U = {Uα : α ∈ Λ} of regular p-closed subsets of X such
that {Uα : α ∈ Λ} = ∅, there exists a finite subset Λ0 of Λ such that
T
{pint Uα : α ∈ Λ0 } = ∅,
(v) for any family
U = {Uα : α ∈ Λ} of pre-regular p-open subsets of X
T
such T
that {Uα : α ∈ Λ} = ∅, there exists a finite subset Λ0 of Λ such
that {Uα : α ∈ Λ0 } = ∅.
Definition 2.9. Let A be a subset of a space X. A point x ∈ X is said
to be a b-pre-θ-accumulation point of A if pcl (U ) ∩ A 6= ∅ for every b-open
subset U of X that contains x. The set of all b-θ-accumulation points of A
is called the b-pre-θ-closure of A and is denoted by b-pclθ (A). A is said to
be b-pre-θ-closed if b-pclθ (A) = A. The complement of a b–pre-θ-closed set
is called b-pre-θ-open.
It is clear that A is b-pre-θ-open if and only if for each x ∈ A, there exists
a b-open set U such that x ∈ U ⊂ pcl U ⊂ A, thus, every b-pre-θ-open set is
b-open.
Definition 2.10.
(i) A space X is called b-regular if for each b-open subset U of X and for
each x ∈ U there exists a b-open subset V of X and a b-closed subset F
of X such that x ∈ V ⊂ F ⊂ U .
(ii) A space X is called strongly b-regular if for each b-open subset U of X
and for each x ∈ U there exists a b-open subset V of X and a preclosed
subset F of X such that x ∈ V ⊂ F ⊂ U .
The following lemma can be easily established.
Lemma 2.11.
(i) A space X is strongly b-regular if and only if every b-open subset of X
is b-pre-θ-open.
(ii) If A is pre-regular p-open, then A is b-pre-θ-closed.
(iii) bcl A ⊂ b-pclθ A.
(iv) If A is preopen, then b-pclθ A = bcl A.
Remark 2.12.
(i) The converse of Lemma 2.11 (ii) is not true, e.g. if X is an infinite set and
τcof is the cofinite topology on X, then in (X, τcof ), every cofinite subset
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M. S. Sarsak
of X is b-pre-θ-open but not pre-regular p-closed as it is not preclosed
(observe that the nonempty b-open (preopen) subsets of (X, τcof ) are
the infinite subsets of X).
(ii) It follows also from Proposition 1.5 that every locally indiscrete space
is strongly b-regular. The converse is, however, not true, e.g. if X is
an infinite set and τcof is the cofinite topology on X, then in (X, τcof ),
every b-open subset of X is b-pre-θ-open. Thus by Proposition 2.11 (i),
X is strongly b-regular. However, (X, τcof ) is not locally indiscrete.
Proposition 2.13. A space X is B-closed if and only if every b-pre-θ-open
cover of X has a finite subcover.
Proof. Necessity. Suppose that X is B-closed and let U = {Uα : α ∈ Λ}
be a b-pre-θ-open cover of X. Then for each x ∈ X, there exists αx ∈ Λ
such that x ∈ Uαx . Since Uαx is b-pre-θ-open, there exists a b-open set
Vx such that x ∈ Vx ⊂ pcl Vx ⊂SUαx , but X is
S B-closed, so there exist
x1 , x2 , . . . , xn ∈ X such that X = ni=1 pcl Vxi = ni=1 Uαxi .
Sufficiency. Follows from Proposition 2.7 and Lemma 2.11 (ii).
Proposition 2.14. Let X be a B-closed, strongly b-regular space. Then
X is finite.
Proof. It follows from Lemma 2.11 (i) and Proposition 2.13, that if X is a
B-closed, strongly b-regular space, then every b-open cover of X has a finite
subcover. Since SO(X) ∪ P O(X) ⊂ BO(X), X is both semi-compact and
strongly compact. Hence, X is finite.
Definition 2.15. A filter base F on a space X is said to b-pre-θ-converge
to a point x ∈ X if for each b-open subset U of X such that x ∈ U , there
exists F ∈ F such that F ⊂ pcl U . F is said to b-pre-θ-accumulate at x ∈ X
if (pcl U ) ∩ F 6= ∅ for every F ∈ F and for every b-open subset U of X such
that x ∈ U .
Observe that if a filter base F b-pre-θ-converges to a point x ∈ X, then
F b-pre-θ-accumulates at x. On the other hand, it is easy to see that a
maximal filter base F b-pre-θ-converges to a point x ∈ X if and only if F
b-pre-θ-accumulates at x.
Proposition 2.16. For a space X, the following are equivalent:
(i) X is B-closed,
(ii) every maximal filter base on X b-pre-θ-converges to some point of X,
(iii) every filter base on X b-pre-θ-accumulates at some point of X.
Proof. (i)→(ii): Let F be a maximal filter base on X such that F does not
b-pre-θ-converge to any point of X. Since F is maximal, F does not b-preθ-accumulate at any point of X. Thus, for each x ∈ X, there exists Fx ∈ F
B-closed spaces
209
and a b-open subset Ux of X such that x ∈ Ux and (pcl Ux ) ∩S
Fx = ∅, but
X is B-closed, so there exist x1 , x2 , . . . , xn ∈ X such that X = ni=1
Tpcl Uxi .
Since F is a filter base on X, there exists F ∈ F such that F ⊂ ni=1 Fxi ,
but (pcl Uxi ) ∩Fxi = ∅S
for each i ∈ {1, 2, . . . , n}, so (pcl Uxi )∩F = ∅ for each
i ∈ {1, 2, . . . , n}, i.e. ( ni=1 pcl Uxi ) ∩ F = X ∩ F = F = ∅, a contradiction.
(ii)→(iii): Let F be a filter base on X. Then F is contained in a maximal
filter base H on X. By (ii), H b-pre-θ-converges to some point x of X, thus
H b-pre-θ-accumulates at x, but F ⊂ H, so F b-pre-θ-accumulates at x.
(iii)→(i): Suppose that X is not B-closed. Then by Proposition 2.8, there
existsTa b-open cover U = {Uα : α ∈ Λ} of X such that for any finite subset Λ0
of
6 ∅. For each finite subset Λ0 of Λ, let FΛ0 =
T Λ, {pint (X\Uα ) : α ∈ Λ0 } =
{pint (X\Uα ) : α ∈ Λ0 }. Then F = {FΛ0 : Λ0 is a finite subset of Λ} is a
filter base on X. Thus by (iii), F b-pre-θ-accumulates at some point x of
X. Since U is a b-open cover of X, there exists α0 ∈ Λ such that x ∈ Uα0 ,
but F b-pre-θ-accumulates at x, so (pcl Uα0 ) ∩ F 6= ∅ for every F ∈ F . Let
F = pint (X\Uα0 ). Then F ∈ F and thus (pcl Uα0 ) ∩ (pint (X\Uα0 )) 6= ∅, a
contradiction.
3. Relatively B-closed subsets
Definition 3.1. A subset A of a space X is called B-closed relative to X
(in X) if any cover of A by b-open subsets of X has a finite subfamily, the
union of the preclosures (in X) of whose members covers A.
Remark 3.2. For the same reason as in Remark 2.2, it is easy to see that
if a subset A of a space X is B-closed in X, then A is p-closed in X and
S-closed in X.
The proof of the following proposition is straightforward and thus omitted.
Proposition 3.3. For a subset A of a space X, the following are equivalent:
(i) A is B-closed in X,
(ii) any cover of A whose members are semi-open or preopen in X has a
finite subfamily, the union of the preclosures (in X) of whose members
covers A,
(iii) any cover of A whose members are regular p-open in X has a finite
subfamily, the union of the preclosures (in X) of whose members covers A,
(iv) any cover of A, whose members are pre-regular p-closed in X, has a
finite subcover of A,
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M. S. Sarsak
(v) for
T any family U = {Uα : α ∈ Λ} of b-closed subsets of X such that
[T {Uα : α ∈ Λ}] ∩ A = ∅, there exists a finite subset Λ0 of Λ such that
[ {pint Uα : α ∈ Λ0 }] ∩ A = ∅,
(vi) for any family U = {Uα : α ∈ Λ}
T each of whose members is semi-closed
or preclosed in X such that [ T
{Uα : α ∈ Λ}] ∩ A = ∅, there exists a
finite subset Λ0 of Λ such that [ {pint Uα : α ∈ Λ0 }] ∩ A = ∅,
(vii) for any
T family U = {Uα : α ∈ Λ} of regular p-closed subsets of X such
that [ T {Uα : α ∈ Λ}] ∩ A = ∅, there exists a finite subset Λ0 of Λ such
that [ {pint Uα : α ∈ Λ0 }] ∩ A = ∅,
(viii) for any family
T U = {Uα : α ∈ Λ} of pre-regular p-open subsets of X
such that [ T
{Uα : α ∈ Λ}] ∩ A = ∅, there exists a finite subset Λ0 of
Λ such that [ {Uα : α ∈ Λ0 }] ∩ A = ∅.
Remark 3.4. Since P RP O(X) = P RP O(X α ), it follows from Proposition
3.3 that a subset A of a space X is B-closed in X if and only if A is B-closed
in X α .
The following proposition is analogous to Proposition 2.13, the similar
proof is omitted.
Proposition 3.5. A subset A of a space X is B-closed in X if and only
if every cover of A by b-pre-θ-open subsets of X has a finite subcover of A.
The following proposition is analogous to Proposition 2.16, the similar
proof is omitted.
Proposition 3.6. For a subset A of a space X, the following are equivalent:
(i) A is B-closed in X,
(ii) every maximal filter base on X that meets A b-pre-θ-converges to some
point of A,
(iii) every filter base on X that meets A b-pre-θ-accumulates at some point
of A.
Proposition 3.7. Let A, B be subsets of a space X. If A is b-pre-θ-closed
and B is B-closed in X, then A ∩ B is B-closed in X.
Proof. Let U = {Uα : α ∈ Λ} be a cover of A ∩ B by b-open sets in X. Then
U ∪ {X\A} is a cover of B. Since X\A is b-pre-θ-open, for each x 6∈ A,
there exists a b-open set Ux such that x ∈ Ux ⊂ pcl Ux ⊂ X\A. Thus
U ∪ {Ux : x ∈ X\A} is a cover of B by b-open sets in X, but B is B-closed
in X, so there
Sn exist α1 , α2 ,S. .m. , αn ∈ Λ and x1 , x2 , . . . , xm ∈ X\A such
that
B
⊂
(
i=1 pcl Uαi ) ∪ ( i=1 pcl Uxi ), but pclUxi ⊂ X\A, so A ∩ B ⊂
Sn
pcl
U
.
αi Hence, A ∩ B is B-closed in X.
i=1
B-closed spaces
211
Corollary 3.8. For a space X, the following are equivalent:
(i) X is B-closed,
(ii) every proper b-pre-θ-closed subset of X is B-closed in X,
(iii) every proper pre-regular p-open subset of X is B-closed in X.
Proof. (i)→(ii): Follows from Proposition 3.7.
(ii)→(iii): Follows from Lemma 2.11 (ii).
(iii)→(i): Let U = {Uα : α ∈ Λ} be a pre-regular p-closed cover of X.
Choose α0 ∈ Λ such that Uα0 6= ∅. Then X\Uα0 is a proper pre-regular
p-open subset of X, thus by (iii), X\Uα0 is B-closed in X,Sso by Proposition
n
3.3, there
Sn exist α1 , α2 , . . . , αn ∈ Λ such that X\Uα0 ⊂ i=1 Uαi and thus
X = i=0 Uαi . Hence by Proposition 2.7, X is B-closed.
In [8], it was pointed out that if A and B are subsets of a space X such
that A ⊂ B and B ∈ SO (X), then pclB A ⊂ pcl A. The following (probably)
known lemma is an improvement of this result, the straightforward proof is
omitted.
Lemma 3.9. Let A and B be subsets of a space X such that A ⊂ B and
B ∈ SO (X). Then
B
(i) IntB A = Int A ∩ B.
(ii) pclB A = (pcl A) ∩ B.
Proposition 3.10. Let A and B be subsets of a space X such that A ⊂ B
and B is α-open. Then A is B-closed in B if and only if A is B-closed in X.
Proof. Necessity. Suppose that A is B-closed in B and let U = {Uα :
α ∈ Λ} be a cover of A by b-open sets in X. Then U B = {Uα ∩ B : α ∈ Λ}
is a cover of A by b-open sets in X and thus in B by Proposition 1.2
(i), but
S A is B-closed in B so there exist α1 , α2 , . . . , αn ∈ Λ such that
A ⊂ ni=1 pclB (Uαi ∩ B). Again by Lemma 3.9 (ii), pclB (Uαi ∩ B) =
(pcl (Uαi ∩ B)) ∩ B ⊂ pcl Uαi . Hence, A is B-closed in X.
Sufficiency. Suppose that A is B-closed in X and let U = {Uα : α ∈ Λ}
be a cover of A by b-open sets in B. Then it follows from Proposition 1.2
(ii) that Uα is b-open in X for each α ∈ Λ,
S but A is B-closed in X, so there
exist α1 , α2 , . . . , αn ∈ Λ such that A ⊂ ni=1 (pcl Uαi ) ∩ B. It follows also
from Lemma 3.9 (ii) that (pcl Uαi ) ∩ B = pclB Uαi . Hence, A is B-closed
in B.
Corollary 3.11. Let A be an α-open subset of a space X. Then A is
B-closed as a subspace if and only if A is B-closed in X.
Corollary 3.12. Let A be a regular open subset of a space X. If X is
B-closed, then A is B-closed as a subspace.
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Proof. Since A is regular open, it is pre-regular p-open, thus by Corollary
3.8, A is B-closed in X, but A is α-open as it is regular open, so it follows
from Corollary 3.11 that A is B-closed as a subspace.
The proof of the following proposition is straightforward and thus omitted.
Proposition 3.13. The finite union of subsets of a space X, each of which
is B-closed in X, is B-closed in X.
Corollary 3.14. If a space X is the finite union of α-open subsets, each
of which is B-closed as a subspace, then X is B-closed.
Proof. Follows immediately from Corollary 3.11 and Proposition 3.13.
Corollary 3.15. ⊕α∈Λ Xα is B-closed if and only if Xα is B-closed for
each α ∈ Λ and Λ is finite.
Proof. Necessity. Since Xα is regular open in ⊕α∈Λ Xα for each α ∈ Λ,
it follows from Corollary 3.12 that Xα is B-closed for each α ∈ Λ. Now
U = {Xα : α ∈ Λ} is a b-open cover of ⊕α∈Λ Xα , but ⊕α∈ΛS
Xα is B-closed,
so there exists a finite subset Λ0 of Λ such that ⊕α∈Λ Xα = α∈Λ0 pcl Xα =
S
α∈Λ0 Xα . Hence, Λ = Λ0 and Λ is finite.
Sufficiency. Follows from Proposition 3.14 since Xα is open and thus
α-open in ⊕α∈Λ Xα for each α ∈ Λ.
Definition 3.16. A function f from a space X into a space Y is called
b-irresolute if f −1 (V ) is b-open in X for every b-open subset V of Y .
It is easy to see that a function f from a space X into a space Y is
b-irresolute if and only if f (bcl A) ⊂ bcl f (A) for every subset A of X. It
follows also from Proposition 1.1 that a function that is both irresolute and
preirresolute is b-irresolute.
Proposition 3.17. Let f be a b-irresolute, preirresolute function from a
space X into a space Y . If A is B-closed in X, then f (A) is B-closed in Y .
Proof. Let U = {Uα : α ∈ Λ}
be a cover of f(A) by b-open subets of Y .
Since f is b-irresolute, V = f −1 (Uα ) : α ∈ Λ is a cover of A by b-open
subsets of X,
α1 , α2 , . . . , αn ∈
Λ such
S but A is B-closed in X, so thereSexist
n
−1 (U ) . Since
that A ⊂ ni=1 pcl f −1 (Uαi ). Thus
f
(A)
⊂
f
pcl
f
αi
i=1
f is preirresolute, f pcl f −1 (Uαi ) ⊂ pcl f f −1 (Uαi ) ⊂ pcl Uαi . Hence,
f (A) is B-closed in Y .
Corollary 3.18. Let f be a b-irresolute, preirresolute function from a
space X onto a space Y . If X is B-closed, then Y is B-closed.
B-closed spaces
213
It is known that homeomorphisms and projection functions are both
semi-irresolute and preirresolute. Thus, the following corollary seems an
immediate consequence of Corollary 3.18.
Corollary 3.19.
(i) The property of being B-closed is topological.
(ii) If ΠXα is B-closed , then Xα is B-closed for each α.
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Received January 1, 2010.