European Journal of Scientific Research
ISSN 1450-216X Vol.47 No.4 (2010), pp.554-561
© EuroJournals Publishing, Inc. 2010
http://www.eurojournals.com/ejsr.htm
On Semiparacompactness and z-paracompactness in
Bitopological Spaces
Luay Abd Al-Haine Al-Swidi
College of Education, Dept. Math., University of Babylon, Babylon, Iraq
Ihsan Jabbar Al-Fatlawe
College of Sciences, Dept. Math., AL-Qadisiyah University, Qadisiyah, Iraq
Summary
We find some properties of semi paracompactness and z-paracompactness in
bitopological spaces and give the relation between these concepts. Throughout the present
paper m will denote infinite cardinal numbers.
Keywords: Paracompact, z-paracompact and bitopological spaces
1. Introduction
The concept of Paracompactness is due to Dieudonne [6] . The concept of paracompact with respect to
three topologies is due to Martin [5] . The term space ( X , τ , µ ) is referred to as a set X with two
generally nonidentical topologies τ and µ .
A cover ( or covering ) of a space ( X , τ ) is a collection of subsets of X whose union is all of
X . A τ -open cover of X is a cover consisting of τ -open sets , and other adjectives applying to subsets
of X apply similarly to covers . If C and ∏ are covers of X , we say ∏ refines C if each members
of ∏ is contained in some member of C . Then, we say ∏ refines ( or is a refinement of ) C . A
collection ∏ of subsets of X is called locally finite if each x in X has a neighborhood meeting only
finitely many member of ∏ , and is called σ -locally finite if it is a countable union of locally finite
collection in X . Note that , every locally finite collection of sets is σ -locally finite . A subset of a
topological space ( X , τ ) is an F σ if it is a countable union of τ - closed sets , and written by τ - F
σ .
1.1. Lemma [6]
Let U be a cover of a topological space X , and let V be a refinement of U . If W refines V , then W
refines U .
1.2. Lemma [6]
Let (Y , τ Y ) be a subspace of ( X , τ ). If a collection V = {Vγ : γ ∈ Γ} of sets is a ( σ -)locally finite
with respect to τ , then so is {Vγ ∩ Y : γ ∈ Γ} with respect to τ Y .
On Semiparacompactness and z-paracompactness in Bitopological Spaces
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1.3. Lemma [6]
1. If U = {U λ : λ ∈ ∆} is locally finite collection of sets in
( X , τ ) . Then any subcollection of U is locally finite .
2. If U = {U λ : λ ∈ ∆} is locally finite collection of sets in
( X , τ ) , then so is {clτ (U λ ) : λ ∈ ∆} and
U clτ (U λ ) = clτ (λUU λ ) .
λ ∈∆
∈∆
3. The union of a finite number of locally finites collections of sets is locally finite.
1.4. Definition [3]
A bitopological space ( X ,τ , µ ) is called pairwise Hausdorff , if for every two distinct points x and y of
X , there exists τ -open set U and µ -open set V such that x ∈ U , y ∈ V and U I V = φ .
1.5. Definition [3]
A bitopological space ( X ,τ , µ ) is (τ ,τ , µ ) -regular , if every point x of X and every τ -open set U
containing x there exists a τ -open set V containing x such that clµ (V ) ⊂ U .
2. Main Results
2.1. Definition
A bitopological space ( X ,τ , µ ) is called (m−)(τ − µ ) semiparacompact with respect to µ [5] , if each
τ -open cover of X ( with cardinality ≤ m ) has a µ -open refinement which is σ -locally finite with
respect to µ .
2.2. Definition
A bitopological space ( X ,τ , µ ) is called (m−)(τ − µ ) -a-paracompact with respect to µ , if each τ open cover of X ( with cardinality ≤ m ) has a refinement which is locally finite with respect to µ .
2.3. Theorem
If ( X ,τ , µ ) is (m−)(τ − µ ) semiparacompact with respect to µ , then the τ -closed subspace
(Y ,τ Y , µY ) is (m)(τ Y − µY ) semiparacompact with respect to µY .
Proof .
Suppose that (Y ,τ Y , µY ) be a τ -closed subspace of ( X ,τ , µ ) . Let U = {U λ : λ ∈ ∆} be a τ Y open cover of Y with cardinality ≤ m . Since each U λ is a τ Y -open subset of Y, there is a τ -open
subset Vλ of X such that U λ = Vλ I Y . Let ∏ = {Vλ : λ ∈ ∆} U { X / Y } . Then ∏ is τ -open cover of X
,( with cardinality ≤ m ) . By hypothesis ∏ has a µ − open refinement W which is σ − locally finite
∞
with respect to µ , hence W = UWn where each Wn = {Wnγ : γ ∈ Γ } is locally finite with respect to µ
n =1
∞
. Let A = U An , where An = {Wnγ ∩ Y : γ ∈ Γ} . We claim that A is
n =1
1. µY − open cover of Y
2. refine U
3. σ − locally finite with respect to µY .
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Luay Abd Al-Haine Al-Swidi and Ihsan Jabbar Al-Fatlawe
is µY − open . Let
is µ − open , then Wnγ I Y
Proof of (1) . Since every Wnγ
y ∈ Y ⇒ y ∈ X ⇒ y ∈ Wnγ for some n , γ , then y ∈ Wnγ I Y for some n , γ . Hence A is a
µY − open cover of Y .
∞
Proof of (2) . Let
U (W γ I Y ) ∈ A
n
where Wnγ I Y = ≠ φ since W refines ∏ , then for every
n =1
∞
∞
∞
UWnγ ∈ W , there is Vλ of ∏ such that
UWnγ ⊂ Vλ , so we get that
UW γ I Y ⊂ Vλ I Y = U λ
n =1
n =1
n =1
n
,
∞
hence
U (W γ I Y ) ⊂ U λ
n
. Therefore A refines U .
n =1
Proof of (3) . By Lemma 1.2 , A is σ − locally finite with respect to µY . Therefore the
subspace (Y ,τ Y , µY ) is a (m)(τ Y − µY ) semiparacompact with respect to µY .
2.4. Theorem
Let ( X ,τ , µ ) be a bitopological space , and let X = { X i : X i ∈ τ I µ , i ∈ I } be a patition of X . The
space ( X ,τ , µ ) is (m−)(τ − µ ) semiparacompact with respect to µ if and only if the space ( X i ,τ i , µi )
is (m−)(τ i − µi ) semiparacompact with respect to µi for every i .
Proof .
The " only if " part , since X i = X / U X j is τ − closed then the subspace ( X ,τ i , µi ) is
j ≠i
(m−)(τ i − µi ) semiparacompact with respect to µi , for every I , by theorem 2.3 .
The"if part" . Let U = {U λ : λ ∈ ∆} be a τ -open cover of X with cardinality ≤ m . The
collection ∏ = {U λ I X i : λ ∈ ∆ } be a τ i − open cover of Xi with cardinal ≤ m for every i .
Since ( X i ,τ i , µi ) is (m−)(τ i − µi ) semiparacompact with respect to µi , for every i , there is a
∞
µi − open refinement Ai which is σ − locally finite with respect to µi so Ai = U Ai , where each
n
n =1
Ain = { Ainγ : γ ∈ Γ } is locally finite with respect to µi .
∞
Let W = UWn where Wn = {U Ainγ : γ ∈ Γ } . We claim that W is
n =1
1. µ -open cover of X .
2. refine U .
3. σ − locally finite with respect to µ .
Proof of (1) . Since Anγ is µi − open , and X i ∈ µ , then Anγ is µ − open . Since
∞
∞
∞
X = U X i = U (U Ai ) = U (U Ai ) = U (U (U Ainγ )) = U (U (U Ainγ )) = U (UWn ) = UW Proof
i∈I
i∈I
i∈I
i∈I
n =1
n =1 i∈I
n =1
of (2)
∞
Let
then
∞
U (U A
UA
n =1
n =1
i nγ ) ∈ W . Since A refine ∏ , then there is a member G of ∏ such that
there
Uλ ∈ U
is
∞
such
UA
i nγ
refine U .
i∈I
n =1
,
⊂G,
hence
∞
⊂ U λ I X i , so U (U Ainγ ) ⊂ U λ I (U X i ) , therefore
n =1
G = Uλ I X i
that
∞
i nγ
i∈I
U (U A
i nγ
n =1 i∈I
) ⊂ U λ I X = U λ . Hence W
On Semiparacompactness and z-paracompactness in Bitopological Spaces
557
Proof of (3). Let x ∈ X , if x ∈ X i , then x has a µi − open neighborhood V such that
V I (U Ainγ ) = φ for all but finitely many γ . Since V is µ − open neighborhood of x , then Wn is
i∈I
locally finite with respect to µ and consequentely W is σ − locally finite with respect to µ . Therefore
( X ,τ , µ ) is (m−)(τ − µ ) semiparacompact with respect to µ .
2.5. Theorem
If ( X ,τ , µ ) is (m−)(τ − µ ) semiparacompact with respect to µ , then the subspace (Y ,τ Y , µY ) is
(m−)(τ Y − µY ) semiparacompact with respect to µY , where Y is τ − Fσ − set .
Proof .
Let U = {U λ : λ ∈ ∆} be a τ Y − open cover of Y ( with cardinality ≤ m ). Since each U λ is
τ Y − open subset of Y then there exists a τ − open set Vλ such that U λ = Vλ I Y .
For each fixed n , ∏ n = {Vλ : λ ∈ ∆ } U { X / Yn } form a τ − open cover of X ( with
cardinality ≤ m ) . By hypothesis ∏ n has a µ − open refinement W which is σ − locally finite with
∞
respect to µ . Then W = UWn where each Wn = { Wnγ : γ ∈ Γ } is locally finite with respect to µ .
n =1
∞
For each n , let B = U Bn , where Bn = {Wnγ I Y : Wn I Y ≠ φ } we claim that B is
n =1
1. µY − open cover of X
2. refines U
3. σ − locally finite with respect to µY .
Proof of (1). Since each Wnγ is µ − open set , then Wnγ I Y is a µY − open set , hence Bn is a
collection of µY − open sets .To show that B covers Y. Let y ∈ Y , then y ∈ Yn for some n , then
y ∈ Wnγ for some γ , then y ∈ Wnγ I Y for some γ , hence B covers Y .
∞
Proof of (2). Let ℑ ∈ B so there exists Wnγ ∈ W such that ℑ = UWnγ I Y . Here Wnγ ⊂ X − Yn
n =1
∞
is impossible, so that Wnγ ⊂ Vλ for some λ , then UWnγ ⊂ Vλ which implase that
n =1
∞
UW γ I Y ⊂ Vλ I Y
n
n =1
, so we get that B ⊂ U λ . Therefore B refines U.
Proof of (3) . By Lemma (1.2) B is σ − locally finite with respect to µY . Therefore the
subspace (Y ,τ Y , µY ) is (m−)(τ Y − µY ) semiparacompact with respect to µY .
2.6. Theorem
Every (m−)(τ − µ ) semiparacompact with respect to µ bitopological space ( X ,τ , µ ) is (m−)(τ − µ ) -aparacompact.
2.7. Definition [ 1 ]
A bitopological space ( X ,τ , µ ) is called (m−)(τ − µ ) compact if for every τ − open cover
U = {U λ : λ ∈ ∆} of X ( with cardinality ≤ m ) has a µ − open finite subcover .
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Luay Abd Al-Haine Al-Swidi and Ihsan Jabbar Al-Fatlawe
2.8. Theorem [1]
If f is a ( µ − τ ' ) closed and ( µ − µ ' ) continuous mapping of a bitopological space ( X ,τ , µ ) onto a
(m _)(τ ' − µ ' ) semiparacompact with respect to µ ' bitopological space (Y ,τ ' , µ ' ) such that z = f −1 ( y ) ,
for all y ∈ Y is (m−)(τ − µ ) compact , then ( X ,τ , µ ) is a (m−)(τ − µ ) semiparacompact with respect to
µ.
2.9. Definition
A bitopolgical space ( X ,τ , µ ) is called (m−)(τ − µ ) semiparacompact with respect to µ , if every
τ − open cover of X (with cardinality ≤ m ) has a µ -closed refinement which is locally finite with
respect to µ .
2.10. Definition
A bitopological space ( X ,τ , µ ) is called (m-)-z- semiparacompact , if every τ − open cover of X (with
cardinality ≤ m ) has a µ -closed refinement which is σ − locally finite with respect to µ .
2.11. Theorem
If a bitopolgical space ( X ,τ , µ ) is a (m−)(τ − µ ) -z-paracompact with respect to µ , then the
τ − closed subspace (Y ,τ Y , µY ) be an (m)(τ Y − µY ) -z-paracompact with respect to µY .
Proof . Let U = {U λ : λ ∈ ∆} be a τ − open cover of X with cardinality ≤ m , then there is a
τ − open subset Vλ of X such that U λ = Vλ I Y for every λ .
The collection ∏ = {Vλ : λ ∈ ∆ } U { X / Y }
form a τ − open cover of X with cardinality ≤ m . Since ( X ,τ , µ ) is a (m−)(τ − µ ) -zparacompact with respect to µ , then ∏ has µ -closed refinement W = { Wγ : γ ∈ Γ } which is locally
finite with respect to µ .
The collection ℘ = { Wγ I Y : γ ∈ Γ } is a µ -closed refinement of U which is locally finite with
respect to µ . Therefore (Y ,τ Y , µY ) is a (m)(τ Y − µY ) -z-paracompact with respect to µY .
2.12. Corollary
If a bitopolgical space ( X ,τ , µ ) is a (τ − µ ) -z-paracompact with respect to µ , then the τ − closed
subspace (Y ,τ Y , µY ) is a (τ Y − µY ) -z-paracompact with respect to µY .
2.13. Theorem
Let ( X ,τ , µ ) be a bitopolgical space and let X = { X i : X i ∈ τ I µ , i ∈ I } be a partition of X .
The bitopolgical space ( X ,τ , µ ) is a (m) (τ − µ ) -z-paracompact with respect to µ if and only if
the space ( X ,τ i , µi ) is a (m) (τ i − µi ) -z-paracompact with respect to µi for every i .
Proof . The "only if " part . Since X = X / U X j is a τ − closed then the subspace ( X ,τ i , µi )
j ≠i
is an m (τ i − µi ) -z-paracompact with respect to µi for every i by Theorem (2.11).
The "if" pat . Let U = {U λ : λ ∈ ∆} be a τ − open cover of X with cardinality ≤ m .The
collection ∏ = { U λ I X i : λ ∈ ∆ } , is a τ i − open cover of X i with cardinality ≤ m for every I . Since
( X ,τ i , µi ) is an m (τ i − µi ) -z-paracompact with respect to µi for every i , there exists a µi − closed
On Semiparacompactness and z-paracompactness in Bitopological Spaces
559
refinement ℜi = { Aiλ : λ ∈ ∆ } of ∏ which is locally finite with respect to µi for every i. Set
W = { U Aiλ : λ ∈ ∆ } .
i∈I
Then W is µ -closed refinement of U which is locally finite with respect to µ . Therefore
( X ,τ , µ ) is an (m) (τ − µ ) -z-paracompact with respect to µ .
2.14. Theorem
If each τ − open in an m (τ − µ ) -z-paracompact with respect to µ bitopological space ( X ,τ , µ ) is an
m (τ − µ ) -z-paracompact with respect to µ , then very subspace (Y ,τ Y , µY ) is an m (τ Y − µY ) -zparacompact with respect to µY .
Proof . Let U = {U λ : λ ∈ ∆} be a τ Y − open cover of Y with cardinality ≤ m . Since each U λ
is τ Y − open in Y , we have U λ = Vλ I Y where Vλ is a τ − open subset of X for every λ ∈ ∆ . Then
G=
UVλ
is a τ − open set .
λ ∈∆
Let V = {Vλ : λ ∈ ∆} be a τ − open cover of G with cardinality ≤ m . Then G is an m (τ − µ ) -zparacompact with respect to µ . Thus V has a µ -closed refinement ℜ = { Aγ : γ ∈ Γ} which is locally
finite with respect to µ . Set ℑ = {Bγ : γ ∈ Γ} , where Bγ = Aγ I Y .
The collection ℑ is µY -closed refinement of U , which is locally finite with respect to µY .
Therefore (Y ,τ Y , µY ) is an m (τ Y − µY ) -z-paracompact with respect to µY .
2.15. Corollary
If each τ − open set in a (τ − µ ) -z-paracompact with respect to µ bitopological space ( X ,τ , µ ) is a
(τ − µ ) -z-paracompact with respect to µ , then every subspace (Y ,τ Y , µY ) is a (τ Y − µY ) -zparacompact with respect to µY .
2.16. Theorem
If ( X ,τ , µ ) be an m (τ − µ ) -z-paracompact with respect to µ ,then the Fσ -subspace (Y ,τ Y , µY ) is an
m (τ Y − µY ) semi-z-paracompact with respect to µY .
Proof . Let U = {U λ : λ ∈ ∆} be a τ Y − open cover of Y with cardinality ≤ m . Since U λ is a
τ Y − open subset of Y for every λ ∈ ∆ , we have U λ = Vλ I Y for every λ ∈ ∆ . For each fixed n ,
En = { Vλ :λ ∈ ∆ } U { X / Yn } form a τ − open cover of X with cardinality ≤ m , since X is an m
(τ − µ ) -z-paracompact with respect to µ , then En has a µ -closed refinement
W = { Wλ : (λ , n) ∈ ∆ × Ν } which is locally finite with respect to µ .
n
For each n , let Bn = { Wλ n I Y : Wλ n I Y ≠ φ } . Then B = U Bn is µ - closed refinement of U
which is σ − locally finite with respect to µ , therefore (Y ,τ Y , µY ) is an m (τ Y − µY ) semi-zparacompact with respect to µY .
2.17. Corollary
If ( X ,τ , µ ) be a (τ − µ ) -z-paracompact with respect to µ ,then the Fσ -subspace (Y ,τ Y , µY ) is a
(τ Y − µY ) semi-z-paracompact with respect to µY .
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Luay Abd Al-Haine Al-Swidi and Ihsan Jabbar Al-Fatlawe
2.18. Theorem
Let ( X ,τ , µ ) be a (τ ,τ , µ ) − regular bitopological space .
If ( X ,τ , µ ) is (τ − µ ) -a-paracompact with respect to µ , then it is (τ − µ ) -z-paracompact with
respect to µ .
Proof. Let U = {U λ : λ ∈ ∆} be a τ − open cover of X . With each x ∈ X , associates a
τ − open set U x containing it and since ( X ,τ , µ ) is (τ ,τ , µ ) − regular , fined a τ − open set Vx with
x ∈ Vx ⊂ clµ (Vx ) ⊂ U x . The collection ∏ = {Vx : x ∈ X } is a τ − open covering since ( X ,τ , µ ) is
(τ − µ ) -a-paracompact with respect to µ , then ∏ has a refinement A = { Ax : x ∈ X } which is locally
finite with respect to µ . The collection C = {clµ ( Ax ) : x ∈ X } is a µ -closed refinement of U which is
locally finite with respect to µ by Lemma (1.3)(2) . Therefore ( X ,τ , µ ) is a (τ − µ ) -z-paracompact
with respect to µ .
2.19. Theorem
Let ( X ,τ , µ ) be a (τ ,τ , µ ) − regular bitopological space .If ( X ,τ , µ ) is (τ − µ ) semiparacompact with
respect to µ , then it is (τ − µ ) -z-paracompact with respect to µ .
Proof . Thus follows from Thorem (2.6) and Theorem (2.18) .
2.20. Definition [2]
A collection of sets U = {U λ : λ ∈ ∆} is said to be conservative ina topological space ( X ,τ ) if
Γ ⊂ ∆ implies that clτ ( U U λ ) =
λ∈Γ
U CLτ (U λ )
.
λ∈Γ
2.21. Proposition [2]
The following statements are equivalent to any collection of sets U = {U λ : λ ∈ ∆}
1. U is conservative ;
2. If Γ ⊂ ∆ , then U cl (U λ ) is τ − closed ;
λ ∈Γ
3. The collection {clτ (U λ ) : λ ∈ ∆ } is conservative.
2.22. Proposition [2]
Every locally finite collection of sets is conservative.
2.23. Theorem
If ( X ,τ , µ ) is m (τ − µ ) -z-paracompact with respect to µ , then every τ − open cover of X with
cardinality ≤ m has a refinement which is a conservative µ − closed cover .
Proof. Let U be a τ − open cover of X . Since ( X ,τ , µ ) is a m (τ − µ ) -z-paracompact with
respect to µ , then U has a µ − closed refinement V which is locally finite with respect to µ . Then by
Proposition (2.22) V is conservative .Thus the result .
2.24. Corollary
If ( X ,τ , µ ) is (τ − µ ) -z-paracompact with respect to µ , then every τ − open cover of X has a
refinement which is a conservative µ − closed cover.
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2.25. Proposition [2]
Let ( X ,τ ) and (Y , µ ) be topological spaces .If f : X → Y a closedmap and U = {U λ : λ ∈ ∆} is a
conservative collection consisting of closed sets in ( X ,τ ) , then ∏ = { f (U λ ) : λ ∈ ∆} is a collection in
(Y , µ ) having the same property .
2.26. Theorem
Let f be a (τ 1 − τ 2 ) continuous and ( µ1 − µ 2 ) closed mapping of a bitopological space ( X ,τ 1 , µ1 ) to a
bitopological space (Y ,τ 2 ,τ 2 ) . If X is a m (τ 1 − µ1 ) -z-paracompact with respect to µ1 , then Y is an m
(τ 2 − µ 2 ) -a-paracompact with respect to µ 2 .
Proof . Let U = {U λ : λ ∈ ∆} be a τ 2 − open cover of Y with cardinality ≤ m . Since f is
(τ 1 − τ 2 ) continuous then ∏ = { f −1 (U λ ) : λ ∈ ∆} will be τ 1 − open cover of X with cardinality ≤ m .
By Theorem (2.23) , U has a refinement V = {Vγ : γ ∈ Γ } which is a conservative µ1 − closed cover .
By Proposition (2.23) , the collection ∏* = { f (Vγ ) : γ ∈ Γ } is a conservative µ 2 − closed cover of Y ,
and is evidently a refinement of U ; hence (Y ,τ 2 ,τ 2 ) is an m (τ 2 − µ 2 ) -a-paracompact with respect to
µ2 .
2.27. Corollary
Let f be a (τ 1 − τ 2 ) continuous and ( µ1 − µ 2 ) closed mapping of a bitopological space ( X ,τ 1 , µ1 ) to a
bitopological space (Y ,τ 2 ,τ 2 ) . If X is a (τ 1 − µ1 ) -z-paracompact with respect to µ1 , then Y is a
(τ 2 − µ 2 ) -a-paracompact with respect to µ 2 .
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