Download Takashi Noiri and Valeriu Popa THE UNIFIED THEORY

DEMONSTRATIO MATHEMATICA
Vol. XLIII
No 1
2010
Takashi Noiri and Valeriu Popa
THE UNIFIED THEORY OF CERTAIN TYPES
OF GENERALIZATIONS OF LINDELÖF SPACES
Abstract. We introduce a new class of sets called ω-m-open sets which are defined
on a family of sets satisfying m-structures with the property of being closed under arbitrary union. The sets enable us to obtain some unified properties of certain types of
generalizations of Lindelöf spaces.
1. Introduction
In 1982, Hdeib [5] introduced the notion of ω-open sets as a weak form of
open sets in topological spaces. By using ω-open sets, he obtained some
improvements of characterizations and preservation theorems of Lindelöf
spaces. Analogously, by using preopen sets Hdeib and Sarsak [7] defined ωpreopen sets and obtained the further properties of strongly Lindelöf spaces
due to Mashhour et al. [11]. As generalizations of Lindelöf spaces, Ganster
[4] introduced and investigated α-Lindelöf spaces and semi-Lindelöf spaces.
Ergun [3] introduced and investigated nearly Lindelöf spaces which have
been further investigated by Balasubramanian [1], Mršević et al. [12] and
Cammaroto and Santoro [2].
In this paper, we introduce the notion of ω-m-open sets in an m-space
(X, mX ) which is equivalent to a generalized topological space defined by
Lugojan [9]. By utilizing ω-m-open sets, we obtain unified characterizations
and preserving theorems for Lindelöf spaces, strongly Lindelöf spaces, αLindelöf spaces, semi-Lindelöf spaces and nearly Lindelöf spaces.
2. Preliminaries
Let (X, τ ) be a topological space and A a subset of X. The closure of
A and the interior of A are denoted by Cl(A) and Int(A), respectively. We
recall some modifications of open sets in topological spaces.
2000 Mathematics Subject Classification: 54A05.
Key words and phrases: m-structure, ω-open, ω-m-open, Lindelöf space.
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T. Noiri, V. Popa
Definition 2.1. Let (X, τ ) be a topological space. A subset A of X is
said to be
(1)
(2)
(3)
(4)
(5)
α-open [13] if A ⊂ Int(Cl(Int(A))),
semi-open [8] if A ⊂ Cl(Int(A)),
preopen [10] if A ⊂ Int(Cl(A)),
regular open if A = Int(Cl(A)),
δ-open [17] if for each x ∈ A, there exists a regular open set U of X such
that x ∈ U ⊂ A, equivalently if A is the union of a family of regular
open sets.
The family of all α-open (resp. semi-open, preopen, δ-open) sets in (X, τ )
is denoted by α(X) (resp. SO(X), PO(X), δ(X)).
Definition 2.2. (1) A topological space (X, τ ) is said to be α-Lindelöf [4]
(resp. semi-Lindelöf [4], strongly Lindelöf [11], nearly Lindelöf [3]) if every
α-open (resp. semi-open, preopen, regular open) cover of X has a countable
subcover.
(2) A subset K of (X, τ ) is said to be α-Lindelöf (resp. semi-Lindelöf ,
strongly Lindelöf [7], nearly Lindelöf ) relative to X if every cover of K by
α-open (resp. semi-open, preopen, regular open) sets of X has a countable
subcover.
Definition 2.3. (1) A topological space (X, τ ) is said to be δ-Lindelöf if
every δ-open cover of X has a countable subcover.
(2) A subset K of (X, τ ) is said to be δ-Lindelöf relative to X if every cover
of K by δ-open sets of X has a countable subcover.
Lemma 2.1. A subset K of a topological space (X, τ ) is δ-Lindelöf relative
to X if and only if K is nearly Lindelöf relative to X.
Proof. Necessity. This is obvious since every regular open set is δ-open.
Sufficiency. Let {Vα : α ∈ Λ} be any cover of K by δ-open sets of X.
For each x ∈ K, there exists α(x) ∈ Λ such that x ∈ Vα(x) . Since Vα(x) is
δ-open, there exists a regular open set Uα(x) such that x ∈ Uα(x) ⊂ Vα(x) .
Since {Uα(x) : x ∈ K} is a cover of K by regular open sets of X and K is
nearly Lindelöf relative to X, there exist countable sets, says, α(x1 ), α(x2 ),
. . . , α(xn ), . . . such that K ⊂ ∪n∈N Uα(xn ) ; hence K ⊂ ∪n∈N Vα(xn ) . This
shows that K is δ-Lindelöf relative to X.
Corollary 2.1. A topological space (X, τ ) is δ-Lindelöf if and only if it
is nearly-Lindelöf .
Proof. This is an immediate consequence of Lemma 2.1.
Definition 2.4. Let (X, τ ) be a topological space. A subset A of X is
said to be ω-open [5] (resp. ω-preopen [7]) if for each x ∈ A, there exists
Generalizations of Lindelöf spaces
205
Ux ∈ τ (resp. Ux ∈ PO(X)) containing x such that Ux \ A is a countable set.
A subset A of X is said to be ω-closed [5] (resp. ω-preclosed [7]) if X \ A is
ω-open (resp. ω-preopen).
For modifications of open sets defined above, the following relationships
are known. Furthermore, it is shown in [7] that the converses need not be
true.
open ⇒ preopen
⇓
⇓
ω-open ⇒ ω-preopen
3. ω-m-open sets
Definition 3.1. Let X be a nonempty set and P(X) the power set of X.
A subfamily mX of P(X) is called a minimal structure (briefly m-structure)
on X if mX satisfies the following properties:
(1) ∅ ∈ mX and X ∈ mX ,
(2) the arbitrary union of the sets belonging to mX belongs to mX .
By (X, mX ), we denote a set X with an m-structure mX and call it an
m-space. Each member of mX is said to be mX -open and the complement
of an mX -open set is said to be mX -closed.
Remark 3.1. (1) An m-structure is equivalent to a generalized topology
due to Lugojan [9].
(2) Let (X, τ ) be a topological space. Then the families α(X), SO(X),
PO(X) and δ(X) are all minimal structures. It is well-known that α(X)
and δ(X) are topologies.
Definition 3.2. Let (X, mX ) be an m-space. A subset A of X is said to
be ω-m-open if for each x ∈ A, there exists Ux ∈ mX containing x such that
Ux \ A is a countable set. The complement of an ω-m-open set is said to be
ω-m-closed. The family of all ω-m-open sets of (X, mX ) is denoted by mω .
Remark 3.2. Let (X, τ ) be a topological space and mX an m-structure
on X.
(1) If mX = τ (resp. PO(X), α(X), SO(X), δ(X)), then an ω-m-open
set is said to be ω-open [5] (resp. ω-preopen [7], ω-α-open, ω-semi-open,
ω-δ-open).
(2) If τ ⊂ mX , then the following relations hold. We can observe that the
implications in the diagram below are not reversible.
open ⇒ m-open
⇓
⇓
ω-open ⇒ ω-m-open
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T. Noiri, V. Popa
Lemma 3.1. A subset A of an m-space (X, mX ) is ω-m-open if and only
if for each x ∈ A, there exists Ux ∈ mX containing x and a countable subset
Cx of X such that Ux \ Cx ⊂ A.
Proof. Necessity. Let A be ω-m-open and x ∈ A. Then there exists
Ux ∈ mX containing x such that Ux \ A is a countable set. Let Cx = Ux \ A.
Then we have Ux \ Cx ⊂ A.
Sufficiency. Let x ∈ A. Then there exists Ux ∈ mX containing x and a
countable set Cx such that Ux \ Cx ⊂ A. Therefore, Ux \ A ⊂ Cx and Ux \ A
is a countable set.
Theorem 3.1. For every m-space (X, mX ), the following properties hold:
(1) the family mω is an m-structure on X and mω is a topology if mX is a
topology,
(2) mX ⊂ mω and (mω )ω = mω .
Proof. (1) It is obvious that ∅, X ∈ mω . Let {Aα : α ∈ Λ} be any subfamily
of mω . Then for each x ∈ ∪α∈Λ Aα , there exists α0 ∈ Λ such that x ∈ Aα0 .
Since Aα0 ∈ mω , there exists Ux ∈ mX containing x such that Ux \ Aα0 is a
countable set. Since Ux \(∪α∈Λ Aα ) ⊂ Ux \Aα0 , Ux \(∪α∈Λ Aα ) is a countable
set. Therefore, ∪α∈Λ Aα is ω-m-open.
Next, suppose that mX is a topology. Let A, B be ω-m-open sets and
x ∈ A ∩ B. Then there exist U, V ∈ mX containing x such that U \ A and
V \ B are countable sets. Since mX is a topology, we have x ∈ U ∩ V ∈ mX .
Furthermore, (U ∩ V ) \ (A ∩ B) = U ∩ V ∩ [(X \ A) ∪ (X \ B)] = [(U ∩ V ∩
(X \ A)] ∪ [U ∩ V ∩ (X \ B)] ⊂ (U \ A) ∪ (V \ B). Therefore, (U ∩ V ) \ (A ∩ B)
is a countable set and hence A ∩ B is ω-m-open.
(2) Since every mX -open set is ω-m-open, by (1) we have mω ⊂ (mω )ω .
Let A ∈ (mω )ω . By Lemma 3.1, for each x ∈ A, there exists Ux ∈ mω
containing x and a countable set Cx such that Ux \ Cx ⊂ A. Furthermore,
by Lemma 3.1, there exists Vx ∈ mX containing x and a countable set Dx
such that Vx \Dx ⊂ Ux . Therefore, we have Vx \(Cx ∪Dx ) = (Vx \Dx )\Cx ⊂
Ux \ Cx ⊂ A. Since Cx ∪ Dx is a countable set, we obtain that A ∈ mω .
Corollary 3.1. (Hdeib [6]) Let (X, τ ) be a topological space. Then τω is
a topology for X finer than τ .
Remark 3.3. Let (X, τ ) be a topological space and mX = δ(X) (resp.
α(X)). Then mω = δω (resp. αω ) is a topology for X finer than δ(X) (resp.
α(X)).
4. m-Lindelöf spaces
Definition 4.1. (1) An m-space (X, mX ) is said to be m-Lindelöf if every
mX -open cover of X has a countable subcover.
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Generalizations of Lindelöf spaces
(2) A subset A of an m-space (X, mX ) is said to be m-Lindelöf relative to mX
if every cover of A by mX -open sets of (X, mX ) has a countable subcover.
Remark 4.1. Let (X, τ ) be a topological space and mX = τ (resp. PO(X),
α(X), SO(X), δ(X)), then an m-Lindelöf space is a Lindelöf space (resp.
strongly Lindelöf space [11], α-Lindelöf space [4], semi-Lindelöf space [4],
nearly Lindelöf space [3]).
Theorem 4.1. A subset K of an m-space (X, mX ) is m-Lindelöf relative
to mX if and only if K is m-Lindelöf relative to mω .
Proof. Necessity. Suppose that a subset K of an m-space (X, mX ) is mLindelöf relative to mX . Let {Uα : α ∈ Λ} be a cover of K by ω-m-open sets
of (X, mX ). For each x ∈ K, there exists α(x) ∈ Λ such that x ∈ Uα(x) . Since
Uα(x) ∈ mω , there exists Vα(x) ∈ mX containing x such that Vα(x) \ Uα(x) is a
countable set. The family {Vα(x) : x ∈ K} is a cover of K by mX -open sets
of (X, mX ). Since K is m-Lindelöf relative to mX , thereSexists a countable
subset, says, α(x1 ), α(x2 ), . . . , α(xn ), . . . such that K ⊂ {Vα(xi ) : i ∈ N}.
Now, we have
K ⊂ ∪i∈N {(Vα(xi ) \ Uα(xi ) ) ∪ Uα(xi ) }
h[
i h[
i
=
(Vα(xi ) \ Uα(xi ) ) ∪
Uα(xi )
i∈N
i∈N
and hence
K⊂
h[
i∈N
i h[
i
(Vα(xi ) \ Uα(xi ) ) ∩ K ∪
Uα(xi ) .
i∈N
For each α(xi ), (Vα(xi ) \ Uα(xi ) ) ∩ K is a countable set and there
S exists a
countable subset Λα(xi ) of Λ such that (Vα(xi ) \ Uα(xi ) ) ∩ K ⊂ {Uα : α ∈
Λα(xi ) }. Therefore, we obtain
i h S
hS
i
∪
{U
:
α
∈
Λ
}
∪
K⊂
U
.
α
α(x
)
α(x
)
i∈N
i∈N
i
i
Sufficiency. Since mX ⊂ mω , the proof is obvious.
Corollary 4.1. An m-space (X, mX ) is m-Lindelöf if and only if (X, mω )
is m-Lindelöf.
Remark 4.2. By Corollary 4.1, we obtain the following result established
in Theorem 4.1 of [6]: a topological space (X, τ ) is Lindelöf if and only if
every ω-open cover of X has a countable subcover.
Theorem 4.2. Let A, B be subsets of an m-space (X, mX ). If A is mLindelöf relative to mX and B is ω-m-closed in (X, mX ), then A ∩ B is
m-Lindelöf relative to mX .
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T. Noiri, V. Popa
Proof. Let B be an ω-m-closed set of (X, mX ). Let {Uα : α ∈ Λ} be a cover
of A ∩ B by mX -open sets of (X, mX ). For each x ∈ A \ B, x ∈ X \ B ∈ mω
and there exists an mX -open set Vx such that Vx ∩ B is a countable set.
Since {Uα : α ∈ Λ} ∪ {Vx : x ∈ A \ B} is a cover of A by mX -open sets
of (X, mX ) and A is m-Lindelöf relative to mX , there exists a countable
subcover {Uα : α ∈ Λ1 } ∪ {Vxi : i ∈ N}, where Λ1 is a countable subset of Λ,
such that
i
i h[
h[
{Vxi : i ∈ N}
{Uα : α ∈ Λ1 } ∪
A⊂
and
A∩B ⊂
h[
i h[
i
{Uα : α ∈ Λ1 } ∪
{B ∩ Vxi : i ∈ N} .
S
Since i∈N (VS
xi ∩ B) is a countable set, there exists a countable subset Λ2 of
Λ such that [ i∈N (Vxi ∩ B)] ⊂ ∪{Uα : α ∈ Λ2 }. Hence {Uα : α ∈ Λ1 ∪ Λ2 } is
a countable subcover of {Uα : α ∈ Λ} and it covers A ∩ B. Therefore, A ∩ B
is m-Lindelöf relative to mX .
Corollary 4.2. Every ω-m-closed subset of an m-Lindelöf space (X, mX )
is m-Lindelöf relative to mX .
Remark 4.3. By Corollary 4.2, we obtain the following results established
in [5] and [7]:
(1) an ω-closed set of a Lindelöf space is Lindelöf (Theorem 3.1(i) of [5]),
(2) an ω-preclosed set of a strongly Lindelöf space is strongly Lindelöf relative
to X (Theorem 2.9 of [7]).
Corollary 4.3. If an m-space (X, mX ) is m-Lindelöf and A is mX -closed
in X, then A is m-Lindelöf relative to mX .
Proof. This is an immediate consequence of Corollary 4.2 since every mX closed set is ω-m-closed.
Corollary 4.4. For an m-space (X, mX ), the following properties are
equivalent:
(1) (X, mX ) is m-Lindelöf;
(2) every proper ω-m-closed set of (X, mX ) is m-Lindelöf relative to mX ;
(3) every proper mX -closed set of (X, mX ) is m-Lindelöf relative to mX .
Proof. (1) ⇒ (2): This is an immediate consequence of Corollary 4.2.
(2) ⇒ (3): Since every mX -closed set is ω-m-closed, the proof is obvious.
(3) ⇒ (1): Let {Vα : α ∈ Λ} be a cover of X by mX -open sets of
(X, mX ). We choose one Vα0 such that X \ Vα0 is a proper subset of X.
Then {Vα : α ∈ Λ \ {α0 }} is a cover of X \ Vα0 by mX -open sets of (X, mX )
and X \ Vα0 is an mX -closed set of X. By (3), there exists a countable
Generalizations of Lindelöf spaces
209
subset Λ0 of Λ such that X \ Vα0 ⊂ ∪{Vα : α ∈ Λ0 }. Therefore, we obtain
X = ∪{Vα : α ∈ Λ0 ∪ {α0 }}. This shows that (X, mX ) is m-Lindelöf.
Remark 4.4. By Corollary 4.4, we obtain the following result established
in Corollary 2.11 of [7]: a topological space (X, τ ) is strongly Lindelöf if and
only if every proper ω-preclosed (preclosed) set is strongly Lindelöf relative
to X.
Definition 4.2. (1) A function f : (X, mX ) → (Y, mY ) is said to be ωm-continuous (resp. M-continuous [16]) if for each x ∈ X and each V ∈ mY
containing f (x), there exists U ∈ mω (resp. U ∈ mX ) containing x such
that f (U ) ⊂ V .
(2) A function f : (X, τ ) → (Y, σ) is said to be ω-continuous [6] if f −1 (V ) is
ω-open in X for every open set V of Y .
Lemma 4.1. For a function f : (X, mX ) → (Y, mY ), the following properties are equivalent:
(1) f is ω-m-continuous;
(2) f : (X, mω ) → (Y, mY ) is M-continuous;
(3) f −1 (V ) ∈ mω for every V ∈ mY .
Proof. This is an immediate consequence of Definition 4.2 and Theorem
3.1.
By Lemma 4.1, we obtain the following result established in [6]: a function f : (X, τ ) → (Y, σ) is ω-continuous if and only if f : (X, τω ) → (Y, σ) is
continuous.
Theorem 4.3. If f : (X, mX ) → (Y, mY ) is an ω-m-continuous function
and K is m-Lindelöf relative to mX , then f(K) is m-Lindelöf relative to mY .
Proof. Let {Vα : α ∈ Λ} be a cover of f (K) by mY -open sets of Y .
Then {f −1 (Vα ) : α ∈ Λ} is a cover of K by ω-m-open sets of X. Since
K is m-Lindelöf relative to mX , by Theorem 4.1 there exists a countable
subset Λ0 of Λ such that K ⊂ ∪{f −1 (Vα ) : α ∈ Λ0 }. Hence we have
f (K) ⊂ ∪{Vα : α ∈ Λ0 }. Therefore, f (K) is m-Lindelöf relative to mY .
Corollary 4.5. If f : (X, mX ) → (Y, mY ) is an ω-m-continuous surjection and (X, mX ) is m-Lindelöf, then (Y, mY ) is m-Lindelöf.
Remark 4.5. If mX = τ , mY = σ and f : (X, mX ) → (Y, mY ) is an ω-mcontinuous surjection, then by Corollary 4.5, we obtain the following result
established in Theorem 4.2 of [6]: if f : (X, τ ) → (Y, σ) is an ω-continuous
surjection and (X, τ ) is a Lindelöf space, then (Y, σ) is Lindelöf.
Corollary 4.6. If f : (X, mX ) → (Y, mY ) is an M-continuous surjection
and (X, mX ) is m-Lindelöf , then (Y, mY ) is m-Lindelöf.
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T. Noiri, V. Popa
A function f : (X, τ ) → (Y, σ) is said to be δ-continuous [14] (resp.
M-precontinuous [11]) if f −1 (V ) is δ-open (resp. preopen) in X for every
δ-open (resp. preopen) set V of Y .
Remark 4.6. By Corollary 4.6, we obtain the following result established in Proposition 7 of [12] (resp. Theorem 5.1 of [11]): if f : (X, τ ) →
(Y, σ) is a δ-continuous (resp. M -precontinuous) surjection and (X, τ ) is
nearly Lindelöf (resp. strongly Lindelöf), then (Y, σ) is nearly Lindelöf (resp.
strongly Lindelöf).
Definition 4.3. A function f : (X, mX ) → (Y, mY ) is said to be ωm-closed (resp. M-closed [15]) if f (A) is ω-m-closed (resp. mY -closed) in
(Y, mY ) for every mX -closed set A of (X, mX ).
Lemma 4.2. For a function f : (X, mX ) → (Y, mY ), the following properties are equivalent:
(1) f is ω-m-closed;
(2) f : (X, mX ) → (Y, mω ) is M-closed;
(3) for each y ∈ Y and each U ∈ mX containing f −1 (y), there exists an
ω-m-open set V of Y containing y such that f −1 (V ) ⊂ U .
Proof. By Theorem 3.1, the proof follows immediately from Theorem 3.1
of [15].
Theorem 4.4. Let f : (X, mX ) → (Y, mY ) be an ω-m-closed function such
that f −1 (y) is m-Lindelöf relative to mX for each y ∈ Y . If K is m-Lindelöf
realtive to mY , then f −1 (K) is m-Lindelöf relative to mX .
Proof. Let {Uα : α ∈ Λ} be any cover of f −1 (K) by mX -open sets of
(X, mX ). For each y ∈ K, f −1 (y) is m-Lindelöf relative to mX and there
exists a countable subset Λ1 (y) of Λ such that f −1 (y) ⊂ ∪{Uα : α ∈ Λ1 (y)}.
Now, we put U (y) = ∪{Uα : α ∈ Λ1 (y)}. Since f is ω-m-closed, by
Lemma 4.2 there exists an ω-m-open set V (y) of Y containing y such that
f −1 (V (y)) ⊂ U (y). Since V (y) is ω-m-open, there exists W (y) ∈ mY containing y such that W (y) \ V (y) is a countable set. For each y ∈ K, we have
W (y) ⊂ (W (y) \ V (y)) ∪ V (y) and hence,
f −1 (W (y)) ⊂ f −1 (W (y) \ V (y)) ∪ f −1 (V (y)) ⊂ f −1 (W (y) \ V (y)) ∪ U (y).
Since W (y) \ V (y) is a countable set and f −1 (z) is m-Lindelöf relative to
mX for each z ∈ Y , there exists a countable set Λ2 (y) of Λ such that
f −1 ([W (y) \ V (y)] ∩ K) ⊂ ∪{Uα : α ∈ Λ2 (y)}
and hence,
f −1 (W (y) ∩ K) ⊂
h[
i h[
i
{Uα : α ∈ Λ2 (y)} ∪
{Uα : α ∈ Λ1 (y)} .
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Generalizations of Lindelöf spaces
Since {W (y) : y ∈ K} is a cover of K by mY -open sets of Y and K is mLindelöf relative to mY , there exist countable points of Y , says, y1 , y2 , . . . , yn ,
. . . such that K ⊂ ∪{W (yi ) : i ∈ N}. Therefore, we obtain
f −1 (K) ⊂ ∪{f −1 (W (yi ) ∩ K) : i ∈ N}
[ h [
⊂
Uα ∪
i∈N
α∈Λ2 (yi )
[
Uα
i
α∈Λ1 (yi )
= ∪{Uα : α ∈ Λ1 (yi ) ∪ Λ2 (yi ), i ∈ N}.
This shows that f −1 (K) is m-Lindelöf relative to mX .
Corollary 4.7. Let f : (X, mX ) → (Y, mY ) be an ω-m-closed function
such that f −1 (y) is m-Lindelöf relative to mX for each y ∈ Y . If (Y, mY ) is
m-Lindelöf, then (X, mX ) is m-Lindelöf.
Corollary 4.8. Let f : (X, mX ) → (Y, mY ) be an ω-m-continuous and
ω-m-closed surjection such that f −1 (y) is m-Lindelöf relative to mX for each
y ∈ Y . Then (X, mX ) is m-Lindelöf if and only if (Y, mY ) is m-Lindelöf.
A function f : (X, τ ) → (Y, σ) is said to be ω-closed [5] (resp. ω-preclosed
[7]) if f (A) is ω-closed (resp. ω-preclosed) in Y for every closed (resp. preclosed) set A of X. By Lemma 4.2, we obtain the following result established
in Theorem 3.1 (ii) of [5]: if f : X → Y is a continuous mapping, then f is
ω-closed if and only if for each y ∈ Y and for each open subset U of X such
that f −1 (y) ⊂ U , there exists an ω-open subset V of Y such that y ∈ V and
f −1 (V ) ⊂ U .
Remark 4.7. By Theorem 4.4 and Corollary 4.7, we obtain the following
results established in Theorem 3.3 of [5] and Theorem 2.20 of [7].
(1) Let f : (X, τ ) → (Y, σ) be a continuous ω-closed surjection such that
f −1 (y) is Lindelöf for each y ∈ Y . If (Y, σ) is Lindelöf, then (X, τ ) is
Lindelöf (Theorem 3.3 of [5]).
(2) Let f : (X, τ ) → (Y, σ) be an ω-preclosed function such that f −1 (y) is
strongly Lindelöf relative to X for each y ∈ Y . If A is a subset of Y which
is strongly Lindelöf relative to Y , then f −1 (A) is strongly Lindelöf relative
to X. In particular, X is strongly Lindelöf whenever Y is (Theorem 2.20
of [7]).
Conclusion. In theorems and corollaries established in this paper, let mX =
τ (resp. PO(X), α(X), SO(X), δ(X)) and mY = σ (resp. PO(Y ), α(Y ),
SO(Y ), δ(Y )), then we can obtain the corresponding properties to Lindelöf
(resp. strongly Lindelöf, α-Lindelöf, semi-Lindelöf, nearly Lindelöf) spaces.
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T. Noiri, V. Popa
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Takashi Noiri
2949-1 SHIOKITA-CHO, HINAGU
YATSUSHIRO-SHI, KUMAMOTO-KEN
869-5142 JAPAN
E-mail:[email protected]
Valeriu Popa
DEPARTMENT OF MATHEMATICS
UNIV. VASILE ALECSANDRI OF BACǍU
600 114 BACǍU, ROMANIA
E-mail:[email protected]
Received February 20, 2009; revised version May 5, 2009.