International Journal of Mathematics and Mathematical Sciences Volume 2007, Article ID 80697,12pages
doi:10.1155/2007/80697
Research Article
On
P3
-Paracompact Spaces
Khalid Al-Zoubi and Samer Al-Ghour
Received 9 April 2007; Accepted 6 August 2007
Recommended by Etienne E. Kerre
Mashhour et al. [1] introduced the notions of P1-paracompactness and P2-paracom pactness of topological spaces in terms of preopen sets. In this paper, we introduce and investigate a weaker form of paracompactness which is calledP3-paracompact. We obtain various characterizations, properties, examples, and counterexamples concerning it and its relationships with other types of spaces. In particular, we show that if a space (X,T) is quasi-submaximal, then (X,T) is paracompact if it isP3-paracompact.
Copyright © 2007 K. Al-Zoubi and S. Al-Ghour. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, dis-tribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Mashhour et al. [1] used preopen sets to define P1-paracompact andP2-paracompact spaces. In [2], Ganster and Reilly studied more properties of such spaces and they proved that these two notions coincide for the class ofT1-spaces.
InSection 2 of the present work, we introduce the notion of p-locally finite collec-tions and study their properties which are used in Section 3 to define the class ofP3 -paracompact spaces. We study these notions in the spaces (X,T), (X,Tα), (X,Tγ), and (X,Ts). Finally, inSection 4, we deal with subspaces, sum, product, images, and the in-verse images ofP3-paracompact spaces.
Throughout this work a space will always mean a topological space on which no sep-aration axiom is assumed unless explicitly stated. Let (X,T) be a space and let Abe a subset ofX. The closure ofA, the interior ofA, and the relative topology onAin (X,T) will be denoted by cl(A), int(A), andTA, respectively.Ais called a preopen subset (see [3]) of (X,T) ifA⊆int(cl(A)). The complement of a preopen set is called a preclosed set. The preclosure ofA, denoted by pcl(A), is the smallest preclosed set that containsA.
(resp.,A=cl(int(A)),A⊆cl(int(A)),A⊆int(cl(int(A)))). The family of all subsets of a space (X,T) which are preopen (resp., preclosed, regular open, regular closed, semiopen) is denoted by PO(X,T) (resp., PC(X,T), RO(X,T), RC(X,T), SO(X,T)). It is known that for a space (X,T) the collection RO(X,T) is a base for a topologyTsonXsuch that
Ts⊆T⊆PO(X,T)⊆PO(X,Ts) and that the collection of allα-sets of (X,T) forms a topology onX, denoted byTα, finer thanTand PO(X,T)=PO(X,Tα).
A space (X,T) is called submaximal [6] if every dense subset of (X,T) is open. It is known that (X,T) is submaximal if and only ifT=PO(X,T). In [7], Al-Nashef intro-duced the notion of quasi-submaximal spaces where a space (X,T) is quasi-submaximal if cl(D)−Dis nowhere dense subset for each dense subsetDof (X,T). This is equivalent to saying that int(D) is dense for each dense subsetDof (X,T) (see [7, Proposition 3.3]). Several characterizations of quasi-submaximal spaces are obtained by Ganster [8, Theo-rems 3.2 and 4.1] and others; see [9,10]. A space (X,T) is called locally indiscrete [11] if every open subset ofXis closed.
For a nonempty setX,Tindisc.andTdis.will denote, respectively, the indiscrete and the discrete topologies onX.
The following lemmas will be used in the sequel.
Lemma 1.1 (see [11]). For a space (X,T), the following items are equivalent: (a) (X,T) is locally indiscrete;
(b) every subset ofXis preopen; (c) every singleton inXis preopen; (d) every closed subset ofXis preopen;
Lemma 1.2 (see [10]). LetAandBbe subsets of a space (X,T). (a) IfA∈PO(X,T) andB∈SO(X,T), thenA∩B∈PO(B,TB). (b) IfA∈PO(B,TB) andB∈PO(X,T), thenA∈PO(X,T).
Lemma 1.3 (see [11]). LetAandBbe subsets of a space (X,T). IfA∈PO(X,T) andB∈T, thenA∩B∈PO(X,T).
2.p-locally finite collections
A collectionᏲ= {Fα:α∈I}of subsets of a space (X,T) is called locally finite (resp., strongly locally finite [12]) if for eachx∈X, there existsUx∈T (resp.,Ux∈RO(X,T)) containingxandUxintersects at most finitely many members ofᏲ.
Definition 2.1. A collectionᏼ= {Pα:α∈I}of subsets of a space (X,T) is calledp-locally finite if for each x∈X, there exists a preopen setWx in (X,T) containingx andWx intersects at most finitely many members ofᏼ.
It follows from the definition andLemma 1.1, if (X,T) is locally indiscrete, then the collectionᏼ= {{x}:x∈X}isp-locally finite.
Application 2.2. LetXbe an infinite set andp∈X.
(a) The space (X,Tindisc.) is locally indiscrete and soᏼ= {{x}:x∈X}is ap-locally finite collection of preopen subsets but not locally finite.
(b) The space (X,T)whereT= {∅,X,{p}}is quasi-submaximal and the collection ᏼ= {{x}:x∈X}isp-locally finite but not locally finite.
Note that if (X,T) is a submaximal space, then every p-locally finite collection of (X,T) is locally finite. However, the converse does not hold in general. LetX= {a,b,c}
with the topologyT= {∅,X,{a,b}}. It is clear that (X,T) satisfies the above property while (X,T) is not submaximal.
Question 2.3. What is the property P on a space (X,T) such that (X,T) satisfies P if and only if everyp-locally finite collection in (X,T) is locally finite?
Theorem 2.4. Letᏼ= {Pα:α∈I}be a collection of semiopen subsets of a space (X,T). Thenᏼisp-locally finite if and only if it is strongly locally finite.
Proof. We need to prove the necessity part only. Letx∈Xand letWxbe a preopen subset of (X,T) such thatx∈WxandWx intersects at most finitely many members ofᏼ, say
Pα1,Pα2,...,Pαn. PutRx=int(cl(Wx)). ThenRx∈RO(X,T) andx∈Rx. We show for every
α∈I− {α1,α2,...,αn},Pα∩Rx= ∅. For eachα∈I, chooseUα∈T such thatUα⊆Pα⊆ cl(Uα). Now, ifPα∩Rx= ∅, then cl(Uα)∩Rx= ∅and soUα∩int(cl(Wx))= ∅which implies that∅ =Uα∩Wx⊆Pα∩Wx. Thus,α∈I− {α1,α2,...,αn}and the result follows.
Corollary 2.5. Every p-locally finite collection of open sets (α-sets, regular closed sets) is locally finite.
Corollary 2.6. (1) A space (X,T) is paracompact (resp., nearly paracompact [13], rc-paracompact [14], S-paracompact [15]) if and only if every open (resp., regular open, regular closed, open) coverᐁofXhas ap-locally finite open (resp., open, regular closed, semiopen) refinement.
(2) A space (X,T) is expandable if and only if for every locally finite collectionᏲ= {Fα:
α∈I}of subsets ofXthere exists ap-locally finite collectionᏼ= {Pα:α∈I}of open subsets ofXsuch thatFα⊆Pαfor eachα∈I.
Theorem 2.7. Letᏼ= {Pα:α∈I}be a collection of subsets of a space (X,T). Then (a)ᏼisp-locally finite if and only if{pcl(Pα) :α∈I}isp-locally finite; (b) ifᏼisp-locally finite, thenα∈Ipcl(Pα)=pcl(α∈IPα);
(c)ᏼis locally finite if and only if the collection{pcl(Pα) :α∈I}is locally finite. The easy proof is left to the reader.
Corollary 2.8. Every p-locally finite collectionᏼ= {Pα:α∈I}of preopen subsets of a quasi-submaximal space (X,T) is locally finite.
Recall that a function f : (X,T)→(Y,M) is called
(a) preirresolute [16] if and only if f−1(A)∈PO(X,T) for eachA∈PO(Y,M);
(b) strongly preclosed [17] iff(A)∈PC(Y,M) for eachA∈PC(X,T).
It is not difficult to see that a function f : (X,T)→(Y,M) is strongly preclosed if and only if for every y∈Y and everyP∈PO(X,T) which contains f−1(y), there is a V∈PO(Y,M) such thaty∈V and f−1(V)⊆P.
Theorem 2.9. Let f : (X,T)→(Y,M) be a preirresolute function. Ifᏼ= {Pα:α∈I}is a
p-locally finite collection in (Y,M), then{f−1(Pα) :α∈I}is a p-locally finite collection in
(X,T).
The proof is obvious.
Recall that a subsetAof a space (X,T) is called strongly compact relative to (X,T) (see [1]) if every cover ofAby preopen sets ofXhas a finite subcover.
Theorem 2.10. Let f : (X,T)→(Y,M) be a strongly preclosed function such that f−1(y)
is strongly compact relative to (X,T) for everyy∈Y. Ifᏼ= {Pα:α∈I}is ap-locally finite collection of subsets of (X,T), then f(ᏼ)= {f(Pα) :α∈I}is ap-locally finite collection in (Y,M).
Proof. Let y∈Y. For eachx∈ f−1(y) chooseP
x∈PO(X,T) such that x∈Px andPx intersects at most finitely many members ofᏼ. The collection{Px:x∈ f−1(y)}is a pre-open cover of f−1(y) and so there exists a finite number of pointsx1,...,xn of f−1(y)
such that f−1(y)⊆n
i=1Pxn=P. Since f is strongly preclosed, there is aV∈PO(Y,M)
such thaty∈Vand f−1(V)⊆P. Therefore,Vintersects at most finitely many members
of f(ᏼ) and thus f(ᏼ) isp-locally finite in (Y,M).
3.P3-paracompact spaces
Recall that a space (X,T) is calledP1-paracompact [1] if every preopen cover ofXhas a locally finite open refinement and it is calledP2-paracompact [1] if every preopen cover ofXhas a locally finite preopen refinement.
Definition 3.1. A space (X,T) is calledP3-paracompact if every open cover ofX has a
p-locally finite preopen refinement.
The following diagram follows immediately from the definitions in which none of the these implications is reversible:
P1-paracompact paracompact P3-paracompact
P2-paracompact
(3.1)
not paracompact since the collection{N{x}:x∈N−} is an open cover ofX which
admits no locally finite open refinement. On the other hand, (X,T) isP3-paracompact since PO(X,T)= {A⊆X:A∩N= ∅}and so the collection{{x}:x∈N}{{x,−x}:
x∈N−}is ap-locally finite preopen cover ofX.
Proposition 3.3. Let (X,T) be a quasi-submaximal space. Then (X,T) is paracompact if and only if it isP3-paracompact.
The proof follows fromCorollary 2.8and the fact that a space (X,T) is paracompact if and only if every open cover ofXhas a locally finite preopen refinement [1].
Example 3.2shows that the condition (X,T) is quasi-submaximal in the above propo-sition is essential.
Recall that a space (X,T) is called countablyP-compact [18], if every countable pre-open cover of (X,T) has a finite subcover. It is clear that everyp-locally finite collection of countablyP-compact space is finite (see [19, proof of Theorem 3.10.3]).
Proposition 3.4. Let (X,T) be a countably P-compact space. Then (X,T) is P3 -para-compact if and only if it is -para-compact.
A space (X,T) is calledp-regular [20] if for each closed setFand each pointx /∈F, there exist disjoint preopen setsUandV such thatx∈UandF⊆V. Note that a space (X,T) is
p-regular if and only if for eachU∈Tand for eachx∈U, there existsP∈PO(X,T) such thatx∈P⊆pcl(P)⊆U.
It is clear that every regular space isp-regular. However, the converse is not true in general as the following example shows.
Application 3.5. LetXbe an infinite set and fixp∈X. ThenT= {U⊆X:p∈UandX−
U is finite {∅} is a topology on Xsuch that PO(X,T)= {S⊆X :p ∈SorSis infinite}{∅}(see [21, Example 1, page 118]). Note that for every open setU∈T con-tainingp, cl(U)=Xand so (X,T) is neither quasi-submaximal nor regular. On the other hand, for everyx∈X, the set{p,x}is both preopen and preclosed and hence (X,T) is
p-regular
Proposition 3.6. Everyp-regular quasi-submaximal space is regular.
Proof. Let (X,T) be a p-regular quasi-submaximal space. LetU∈T andx∈U. Since (X,T) isp-regular, there existsV∈PO(X,T) such thatx∈V⊆pcl(V)⊆U. Since (X,T) is quasi-submaximal and V ∈PO(X,T), cl(V)=pcl(V). Therefore, x∈int(cl(V))⊆
cl(int(cl(V)))=cl(V)=pcl(V)⊆Uand hence (X,T) is regular.
Proposition 3.7. EveryP3-paracompactT2-space (X,T) isp-regular.
Proof. LetAbe a closed subset of (X,T) and letx /∈A. For eachy∈A, choose an open setUysuch thaty∈Uyandx /∈cl(Uy). Therefore, the familyᐁ= {Uy:y∈A}{X−A} is an open cover of (X,T) and so it has ap-locally finite preopen refinementᏴ. PutV=
{H∈Ᏼ:H∩A= ∅}. ThenV is a preopen set containingAand pcl(V)={pcl(H) :
Theorem 3.8. Let (X,T) be a regular space. Then (X,T) isP3-paracompact if and only if every open coverᐁofXhas ap-locally finite preclosed refinementᐂ(that isV∈PC(X,T) for everyV∈ᐂ).
Proof. To prove necessity, letᐁbe an open cover ofX. For eachx∈Xwe choose a mem-berUx∈ᐁand, by the regularity of (X,T), an open subsetVx∈T such thatx∈Vx⊆ cl(Vx)⊆Ux. Therefore,ᐂ= {Vx:x∈X}is an open cover ofX and so by assumption, it has a p-locally finite preopen refinement, sayᐃ= {Wβ:β∈B}. Now, consider the collection pcl(ᐃ)= {pcl(Wβ) :β∈B}. It is easy to see that pcl(ᐃ) is a p-locally finite collection (Theorem 2.7(a)) of preclosed subsets of (X,T) such that for every β∈B, pcl(Wβ)⊆pcl(Vx)⊆cl(Vx)⊆Ux for someUx∈ᐁ, that is, pcl(ᐃ) is a refinement of ᐁ.
To prove sufficiency, letᐁbe an open cover ofXand letᐂbe a p-locally finite pre-closed refinement ofᐁ. For eachx∈X, chooseWx∈PO(X,T) such thatx∈Wx and
Wxintersect at most finitely many members ofᐂ. LetᏴbe a preclosed p-locally finite refinement ofᐃ= {Wx:x∈X}. For eachV ∈ᐂ, letV =X− {H∈Ᏼ:H∩V = ∅}. Then{V :V ∈ᐂ} is a preopen cover ofX. Finally, for eachV ∈ᐂ, chooseUV ∈ᐁ such thatV⊆UV.Therefore, the collection{UV∩V :V∈ᐂ}is a locally finite preopen (Lemma 1.3) refinement ofᐁand thus (X,T) isP3-paracompact.
Observe that to prove the sufficiency part ofTheorem 3.8, there is no need for the space (X,T) to be regular. However, the condition (X,T) that is regular in the necessity part cannot be replaced byp-regular asExample 3.5shows.
Recall that a subsetAof a space (X,T) is called aγ-set [22] ifA∩P∈PO(X,T) for eachP∈PO(X,T). It is well known that the collection of allγ-sets in a space (X,T), denoted byTγ, is a topology onX satisfyingT⊆Tα⊆Tγ⊆PO(X,Tγ)⊆PO(X,Tα)= PO(X,T).
Ganster [8, Theorems 3.2 and 4.1] shows that if (X,T) is quasi-submaximal, thenTα=
PO(X,T).
Proposition 3.9. Let (X,T) be a space.
(1) If (X,T) isP1-paracompact, then (X,Tα) isP1-paracompact. (2) (X,T) isP2-paracompact if and only if (X,Tα) isP2-paracompact. (3) If (X,Tα) isP3-paracompact, then (X,T) isP3-paracompact.
Proof. The proofs of (1) and (3) are clear. To prove (2), we need to prove the sufficiency part only. Letᐁbe a preopen cover of (X,T). Since PO(X,Tα)=PO(X,T),ᐁis a pre-open cover of theP2-paracompact space (X,Tα) and so it has a locally finite preopen refinement ᐂin (X,Tα). As in the proof of [15, Theorem 2.10],ᐂis locally finite in (X,T). Therefore,ᐂis a locally finite preopen refinement ofᐁin (X,T) and thus (X,T) isP2-paracompact.
The following examples show that the converses of part (1) and part (3) ofProposition
3.9are not true in general.
collection{{p,x}:x∈X}is an open (preopen) cover of (X,Tα) which admits no locally finite preopen refinement in (X,Tα).
(b) Let X = {a,b,c} with the topology T= {∅,X,{a}}. Then Tα=PO(X,T)= PO(X,Tα)= {∅,X,{a},{a,b},{a,c}}and so (X,Tα) isP1-paracompact but (X,T) is not.
Proposition 3.11. Let (X,T) be a space. Then
(1) if (X,T) isP1-paracompact, then (X,Tγ) isP1-paracompact; (2) if (X,T) isP2-paracompact, then (X,Tγ) isP2-paracompact; (3) if (X,Tγ) isP3-paracompact, then (X,T) isP3-paracompact.
Proof. The proofs of (1) and (3) are clear. For (2), letᐁbe a preopen cover of (X,Tγ). Since PO(X,Tγ)⊆PO(X,T),ᐁis a preopen cover of theP2-paracompact space (X,T) and so it has a locally finite preopen refinement ᐂin (X,T). Now, for everyV ∈ᐂ, chooseUV∈ᐁsuch thatV⊆UV. Then, byLemma 1.3, the collection{int(cl(V)∩UV:
V∈ᐂ}is a locally finite preopen refinement ofᐁin (X,Tγ).
The converse of each part of Proposition 3.11 is not true in general. For (1), see Example 3.10(a) and for (2), seeExample 2.2(a). Finally, for (3), we consider the space (X,T) given inExample 3.10(a). Then (X,T) is a quasi-submaximal semi-TDspace (see [2]) and soTα=Tγ=PO(X,T)=PO(X,Tγ) (see [21, page 117]). Therefore, (X,T) is
P3-paracompact but (X,Tγ) is not.
Proposition 3.12. Let (X,T) be a space. Then
(1) if (X,Ts) isP1-paracompact, then (X,T) isP1-paracompact; (2) if (X,Ts) isP2-paracompact, then (X,T) isP2-paracompact; (3) if (X,T) isP3-paracompact, then (X,Ts) isP3-paracompact.
Proof. The proofs of (1) and (3) are clear. The proof of (2) is similar to the proof of Proposition 3.11(2).
To see that the converse of (1) is not true in general, we consider the space (X,T), where X= {a,b} and T= {∅,X,{a}}. ThenT =PO(X,T) and so (X,T) is P1 -para-compact while (X,Ts) is not sinceTs=Tindisc..
To see that the converse of (3) is not true in general, we consider the following example.
Application 3.13. LetXbe an infinite set and let p∈X with the topologyT= {U⊆X:
p∈U}{∅}. ThenTs=Tindisc.and so (X,Ts) isP3-paracompact. However, the space (X,T) is not since the collection {{p,x}:x∈X}is an open (preopen) cover of (X,T) which admits nop-locally finite preopen refinement in (X,T).
4. Operations
Definition 4.1. A subsetAof a space (X,T) is calledP3-paracompact relative to (X,T) if every cover ofAby open subsets of (X,T) has a p-locally finite preopen refinement in (X,T).
(resp., U∈PO(X,T)) such that x∈U⊆cl(U)⊆A (resp.,x∈U⊆pcl(U)⊆A). The complement of aθ-open (resp., pre-θ-open) set is calledθ-closed (resp., pre-θ-closed).
It is clear that everyθ-open is pre-θ-open. However, the converse is not true in general. Since in (X,Tindisc.), every subset of (X,Tindisc.) is pre-θ-open .
Lemma 4.2. LetAbe a subset of a quasi-submaximal space (X,T). ThenAis pre-θ-open if and only if it isθ-open.
Proof. We need to prove the necessity part only. Letx∈A. ChooseU∈PO(X,T) such thatx∈U⊆pcl(U)⊆A. Since (X,T) is quasi-submaximal pcl(U)=cl(U) and cl(U)∈
RC(X,T). Therefore,x∈int(cl(U))⊆cl(int(cl(U)))=cl(U)=pcl(U)⊆Aand thusAis
θ-open.
Theorem 4.3. LetAbe aP3-paracompact relative subset of aT2-space (X,T). ThenAis pre-θ-closed.
Proof. Letx /∈A. For each y∈A, there existsUy∈T such thaty∈Uyandx /∈cl(Uy). Therefore, the familyᐁ= {Uy:y∈A}is an open cover ofAin (X,T). SinceAisP3 -paracompact relative to (X,T),ᐁhas ap-locally finite preopen refinement in (X,T), say ᐂ. PutW={V :V∈ᐂ}andW =X−pcl(W). ThenW ∈PO(X,T) and sinceT⊆
PO(X,T), x∈W. Moreover, byTheorem 2.7(b), pcl(W )∩A=∅, that is, x∈W ⊆
pcl(W )⊆X−A. This shows thatX−Ais pre-θ-open and henceAis pre-θ-closed.
Theorem 4.4. LetAbe ag-closed subset of aP3-paracompact space (X,T). ThenAisP3 -paracompact relative to (X,T).
Proof. Letᐁ= {Uα:α∈I}be a cover ofAby open subsets of (X,T). SinceA⊆{Uα:
α∈I}andAisg-closed, cl(A)⊆{Uα:α∈I}. For eachx /∈cl(A), there exists an open setWx of (X,T) such thatA∩Wx= ∅. Now, putᐁ = {Uα:α∈I}{Wx:x /∈cl(A)}. Thenᐁ is an open cover of theP3-paracompact space (X,T). LetᏴ= {Hβ:β∈B}be a p-locally finite preopen refinement ofᐁ and putᏴᐁ= {Hβ:Hβ⊆Uα(β),β∈Band α(β)∈I}. ThenᏴᐁ is a p-locally finite preopen refinement ofᐁ. Therefore,A isP3
-paracompact relative to (X,T).
Corollary 4.5. Let (X,T) be a quasi-submaximalP3-paracompactT2-space andAany subset ofX. Then, the following items are equivalent:
(a)AisP3-paracompact relative to (X,T); (b)Ais pre-θ-closed;
(c)Aisθ-closed; (d)Ais closed; (e)Aisg-closed.
At this point it is important to give an example of a quasi-submaximalT2-space which is not submaximal.
other hand, D∗=D{x},x /∈D is a nonopen dense subset of (X,T∗) and therefore (X,T∗) is not submaximal.
Theorem 4.7. LetA be a regular closed subset of aP3-paracompact space (X,T). Then (A,TA) isP3-paracompact.
Proof. Letᐂ= {Vα:α∈I}be an open cover ofAin (A,TA). For eachα∈I, chooseUα∈
Tsuch thatVα=A∩Uα. Then the collectionᐁ= {Uα:α∈I}{X−A}is an open cover of theP3-paracompact space (X,T) and so it has a p-locally finite preopen refinement, sayᐃ= {Wβ:β∈J}. Then, byLemma 1.2(a) and the fact that RC(X,T)⊆SO(X,T), the collection{A∩Wβ:β∈J}is a p-locally finite preopen refinement ofᐂin (A,TA).
This completes the proof.
Theorem 4.8. LetAandBbe subsets of a space (X,T) such thatA⊆B.
(a) IfB∈PO(X,T) andAisP3-paracompact relative to (B,TB), thenAisP3 -para-compact relative to (X,T).
(b) IfB∈SO(X,T) andAis P3-paracompact relative to (X,T), thenAisP3 -para-compact relative to (B,TB).
Proof. (a) Let ᐁ= {Uα:α∈I} be an open cover of Ain (X,T). Then the collection ᐁB= {B∩Uα:α∈I}is an open cover ofAin (B,TB). SinceAisP3-paracompact relative to (B,TB),ᐁBhas ap-locally finite preopen refinementᐂᏮin (B,TB). ByLemma 1.2(b),
the collection ᐂB is a p-locally finite preopen refinement of in (X,T). Therefore,A is
P3-paracompact relative to (X,T).
(b) Letᐂ= {Vα:α∈I}be an open cover ofAin (B,TB). For everyα∈I, choose
Uα∈T such thatVα=B∩Uα. Therefore, the collectionᐁ= {Uα:α∈I} is an open cover ofAin (X,T) and so it has a p-locally finite preopen refinement ofᐁin (X,T), sayᏼ. Then, byLemma 1.2(a), the collectionᏼB= {P∩B:P∈ᏼ}is ap-locally finite
preopen refinement ofᐁin (B,TB).
Corollary 4.9. LetAa subset of a space (X,T).
(a) IfA∈PO(X,T) and the subspace (A,TA) isP3-paracompact, thenAisP3 -para-compact relative to (X,T).
(b) IfA∈SO(X,T) and A is P3-paracompact relative to (X,T), then the subspace (A,TA) isP3-paracompact.
(c) IfA∈Tα, thenAisP3-paracompact relative to (X,T) if and only if the subspace (A,TA) isP3-paracompact.
InExample 3.2, if we putA=N−{1}. ThenA /∈SO(X,T) andTA= {B{1}:B⊆
N−}{∅}. Since the collection {{1}{{n,−n}:n∈N}is a p-locally finite preopen
cover ofAin (X,T),AisP3-paracompact relative to (X,T). On the other hand, for ev-eryn∈N,{1,−n} ∈TA and{−n}∈/ PO(A,TA) and so the subspace (A,TA) is notP3 -paracompact. Therefore, inCorollary 4.9(a), the conditionAthat is semiopen cannot be dropped.
To see that the conditionAis preopen inCorollary 4.9(a) is essential, we consider the space (X,T) given inExample 3.13. Now, putA=X− {p}. ThenA /∈PO(X,T) andTA=
to (X,T) since the collection{{a,p}:a∈A}is an open cover ofAin (X,T) which admits nop-locally finite preopen refinement in (X,T).
Proposition 4.10. LetAbe a pg-closed subset of a space (X,T) and letBbe any subset of
X. IfAisP3-paracompact relative to (X,T) andA⊆B⊆pcl(A), thenBisP3-paracompact relative to (X,T).
The easy proof is left to the reader.
Recall that a functionf : (X,T)→(Y,M) is called M-preopen [1] if f(A)∈PO(Y,M) for eachA∈PO(X,T).
It is clear that every continuous open function is preirresolute and M-preopen (see [19, Exercise 1.4.C]).
Theorem 4.11. Let f : (X,T)→(Y,M) be a continuous, open, and strongly preclosed sur-jective function such that f−1(y) is strongly compact relative to (X,T) for everyy∈Y. If
(X,T) isP3-paracompact, then so is (Y,M).
Proof. Letᐂ= {Vα:α∈I}be an open cover of (Y,M). Since f is continuous, the col-lectionᐁ= f−1(ᐂ)= {f−1(Vα) :α∈I}is an open cover of theP3-paracompact (X,T)
space and so it has ap-locally finite preopen refinement, sayᏼ. Sincef is continuous and open, thenf is M-preopen and so byTheorem 2.10, the collectionf(ᏼ) is ap-locally fi-nite preopen refinement ofᐂ. The result now follows.
Theorem 4.12. Let f : (X,T)→(Y,M) be a closed preirresolute surjective function such that f−1(y) is compact in (X,T) for every y∈Y. If (Y,M) isP3-paracompact, then so is
(X,T).
Proof. Letᐁ= {Uα:α∈I}be an open cover of space (X,T). For each y∈Y,ᐁis an open cover of the compact subspace f−1(y) and so there exists a finite subsetI(y) ofI
such that f−1(y)⊆
α∈I(y)Uα=UyandUyis open in (X,T). As f is a closed function,
for eachy∈Ywe find an open subsetVyofYsuch thaty∈Vyand f−1(y)⊆Uy. Then the collection ᐂ= {Vy:y∈Y} is an open cover of theP3-paracompact space (Y,M) and so it has a p-locally finite preopen refinement, say ᐃ= {Wβ:β∈B}. Since f is preirresolute, the family f−1(ᐃ)= {f−1(Wβ) :β∈B}is a preopenp-locally finite cover
of (X,T) such that for eachβ∈B,f−1(Wβ)⊆Uyfor somey∈Y. Finally, the collection {f−1(Wβ)∩U
α(x):β∈B, α(x)∈I(y)}is a p-locally finite preopen refinement of ᐁ.
Therefore, (X,T) isP3-paracompact.
Corollary 4.13. The product space of a compact space and a P3-paracompact space is
P3-paracompact.
Theorem 4.14. The topological sum ⊕α∈IXα isP3-paracompact if and only if the space (Xα,Tα) isP3-paracompact, for eachα∈I.
Putᐂ=α∈Iᐂα. It is clear that ᐂis a p-locally finite refinement ofᐁsuch thatV∈ PO(X,T) for everyV∈ᐂ(Lemma 1.2(b)). Thus⊕α∈IXαisP3-paracompact.
Acknowledgment
The publication of this paper was supported by Yarmouk University Research Council.
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Khalid Al-Zoubi: Department of Mathematics, Faculty of Science, Yarmouk University, Irbid 21163, Jordan
Email address:[email protected]
Samer Al-Ghour: Department of Mathematics and Statistics, Faculty of Science, Jordan University of Science and Technology, Irbid 22110, Jordan
