© Hindawi Publishing Corp.
ON p-CLOSED SPACES
JULIAN DONTCHEV, MAXIMILIAN GANSTER, and TAKASHI NOIRI (Received 31 December 1998)
Abstract.We will continue the study ofp-closed spaces. This class of spaces is strictly placed between the class of strongly compact spaces and the class of quasi-H-closed spaces. We will provide new characterizations ofp-closed spaces and investigate their relationships with some other classes of topological spaces.
Keywords and phrases. p-closed, QHC, strongly compact, nearly compact, preopen, con- solidation.
2000 Mathematics Subject Classification. Primary 54D20, 54D25; Secondary 54A05, 54D30, 54H05.
1. Introduction and preliminaries. The aim of this paper is to continue the study ofp-closed spaces, which were introduced by Abo-Khadra [1]. A topological space (X,τ) is called p-closed if every preopen cover ofX has a finite subfamily whose pre-closures coverX.
LetAbe a subset of a topological space(X,τ). Following Kronheimer [13], we call the interior of the closure ofA, denoted byA+, theconsolidationofA. Sets included in their consolidation play a significant role in, e.g., questions concerning covering properties, decompositions of continuity, etc. Such sets are calledpreopen[15] orlocally dense[4].
A subset A of a space(X,τ) is calledpreclosed if its complement is preopen, i.e., if cl(intA)⊆A. The preclosure ofA⊆X, denoted by pcl(A), is the intersection of all preclosed supersets ofA. Since any union of preopen sets is also preopen, the preclosure of every set is preclosed. It is well known that pclA=A∪cl(intA)for any A⊆X.
Another interesting property of preopen sets is the following: when a certain topo- logical property is inherited by both open and dense sets, it is often then inherited by preopen sets.
Several important concepts in topology are and can be defined in terms of pre- open sets. Among the most well known are Bourbaki’s submaximal spaces (see [2]).
A topological space is calledsubmaximalif every (locally) dense subset is open or, equivalently, if every subset is locally closed, i.e., the intersection of an open set and a closed set. Another class of spaces commonly characterized in terms of preopen sets is the class of strongly irresolvable spaces introduced by Foran and Liebnitz in [9]. A topological space(X,τ)is calledstrongly irresolvable[9] if every open subspace ofX is irresolvable, i.e., it cannot be represented as the disjoint union of two dense subsets.
Subspaces that contain two disjoint dense subsets are calledresolvable. Ganster [10]
has pointed out that a space is strongly irresolvable if and only if every preopen set
is semi-open, where a subsetSof a space(X,τ)is calledsemi-openifS⊆cl(intS). We will denote the families of preopen (respectively, semi-open) sets of a space(X,τ)by PO(X)(respectively, SO(X)).
Many classical topological notions such as compactness and connectedness have been extended by using preopen sets instead of open sets. Among them are the class ofstrongly compact spaces[16] (= every preopen cover has a finite subcover) stud- ied by Jankovi´c, Reilly and Vamanamurthy [12] and by Ganster [11], and the class of preconnected spaces(= spaces that cannot be represented as the disjoint union of two preopen subsets) introduced by Popa [19]. The study of topological properties via preopenness has gained significant importance in general topology and one example for that is the fact that four (out of the ten) articles in the 1998 Volume of “Memoirs of the Faculty of Science Kochi University Series A Mathematics” were more or less devoted to preopen sets.
A pointx∈Xis called aδ-cluster pointof a setA[25] ifA∩U= ∅for every regular open setUcontainingx. The set of allδ-cluster points ofAforms theδ-closureofA denoted by clδ(A), andAis calledδ-closed[25] ifA=clδ(A). IfA⊆int(clδ(A)), thenA is said to beδ-preopen[21]. Complements ofδ-preopen sets are calledδ-preclosedand theδ-preclosureof a setA, denoted byδ-pcl(A), is the intersection of allδ-preclosed supersets ofA.
Following [22], we will call a topological space(X,τ) δp-closedif for everyδ-preopen cover{Vα:α∈A} ofX, there exists a finite subsetA0of A such thatX = ∪{δ− pcl(Vα):α∈A0}.
2. p-closed spaces
Definition2.1. A topological space(X,τ)is said to bep-closed[1] (respectively, quasi-H-closed=QHC) if for every preopen (respectively, open) cover{Vα:α∈A}of X, there exists a finite subsetA0ofAsuch thatX= ∪{pcl(Vα):α∈A0}(respectively, X= ∪{cl(Vα):α∈A0}).
It is clear that every strongly compact space isp-closed, and that every p-closed space is QHC. We also observe that a space(X,τ)is QHC if and only if every preopen cover has a finite dense subsystem (=finite subfamily whose union is a dense subset).
Since every preopen set isδ-preopen, we haveδ-pclS⊆pclS for everyS ⊆X. This implies that everyδp-closed space isp-closed.
Theorem2.2. Let(X,τ)be QHC and strongly irresolvable. Then(X,τ)is p-closed.
Proof. Let{Si:i∈I}be any preopen cover ofX. SinceXis QHC, there exists a finite subsetAofI such thatX= ∪{cl(Si):i∈A}. SinceXis strongly irresolvable, Si∈SO(X)and therefore cl(Si)=cl(int(Si))=pcl(Si)for each i∈I. HenceXis p- closed.
Corollary2.3. Let(X,τ)be strongly irresolvable (or submaximal). Then(X,τ)is p-closed if and only if it is QHC.
Observe that ap-closed space need not be strongly irresolvable as any finite indis- crete space shows. However, we do have the following result.
Theorem2.4. Let(X,τ)be ap-closedT0space. Then(X,τ)is strongly irresolvable.
Proof. Suppose thatWis a nonempty, open, and resolvable subspace ofX. Then Wis dense-in-itself and also infinite, since(X,τ)isT0. LetW=E1∪E2, whereE1and E2are disjoint dense subsets ofW, and, without loss of generality, we may assume thatE1is infinite. Moreover, letA= {x∈E1:{x} ∈PO(X)}. Observe that for each y∈E1\A, {y}is nowhere dense. Now picky∈E1\A. IfSy =(X\W )∪E2∪ {y}
thenSy is dense and therefore preopen. IfGis a nonempty open set contained inSy, thenG∩E1⊆ {y}and soG∩W⊆cl(E1)⊆cl{y}. Since{y}is nowhere dense,G∩W is empty and so cl(int(Sy))⊆X\W, thus pclSy=Sy. Now, observe that{{x}:x∈ A}∪{Sy:y∈E1\A}is a preopen cover ofX. Hence there exists a finite subsetA1of Aand a finite subsetA2ofE1\Asuch thatX= {{x}:x∈A1}∪{Sy:y∈A2}. Then, E1⊆A1∪A2which is a contradiction. ThusXis strongly irresolvable.
By combining the previous two results we immediately have the following theorem.
Theorem2.5. Let(X,τ)be aT0space. Then(X,τ)isp-closed if and only if(X,τ) is QHC and strongly irresolvable.
The following diagram exhibits the relationships between the class of p-closed spaces and some related classes of topological spaces. Note that none of the implica- tions is reversible
strongly compact
//
p-closed
,, X
X X X X X X X X X X X X X X X X X X X X X X
X X
δp-closedoo
α-compact
//
compact//
nearly compact//
QHC.semi-compact
OO //
s-closed// 55 l l l l l l l l l l
S-closed
66 m
m m m m m m m m m
(2.1)
Example2.6. (i) Recall that a space(X,τ)is calledα-scattered[7] if it has a dense set of isolated points. Clearly everyα-scattered space is strongly irresolvable and so, by Theorem 2.2, everyα-scattered QHC space isp-closed. In particular, the Katetov extensionκNof the set of natural numbersN(cf. [20]) isp-closed and not compact, hence not strongly compact.
(ii) The unit interval[0,1]with the usual topology is compact, hence QHC, but not p-closed since it is resolvable.
(iii) LetX=R,τ= {∅,{0},X}. Then,Xisp-closed ands-closed but notα-compact and hence not strongly compact (a space isα-compactif every cover byα-open sets has a finite subcover, where a set isα-openif it is the difference of an open and a nowhere dense set; clearly everyα-open set is preopen but not vice versa). Additionally, this space is notδp-closed since every subset isδ-preopen.
We next discuss the relationship betweenp-closedness and compactness. Recall that a space(X,τ)is callednearly compact[24] if every cover ofX by regular open sets has a finite subcover, i.e., the semiregularization(X,τs)of (X,τ)is compact.
Example 4.8(d) in [20] shows that there exists a Hausdorff, non-compact, semi-regular and QHC space with a dense set of isolated points. Such a space isp-closed but not nearly compact. Example 2.10 in [22] provides another such example.
For any infinite cardinalκ, a topological space(X,τ)is calledκ-extremally discon- nected(=κ-e.d.) [6] if the boundary of every regular open set has cardinality (strictly) less thanκ. Several topological spaces share this property forκ= ℵ0. Since there are finite spaces which fail to be extremally disconnected, clearlyℵ0-extremal disconnect- edness is a strictly weaker property than extremal disconnectedness.
Theorem2.7. If a topological space (X,τ)is p-closed and ℵ0-extremally discon- nected (respectively, extremally disconnected), then(X,τ)is nearly compact (respec- tively,s-closed).
Proof. We first prove the case when the space isℵ0-extremally disconnected. Let {Ai:i∈I}be any regular open cover ofX. For eachi∈I, we have pcl(Ai)=Ai∪ cl(int(Ai))=cl(Ai). SinceX is p-closed, then there exists a finite F ⊆I such that X= ∪i∈Fcl(Ai). Note that for eachAi, we have cl(Ai)=Bi∪Ci, whereBi=int(cl(Ai)) andCi=cl(Ai)\int(cl(Ai)). SinceX isℵ0-extremally disconnected, thenCi is finite for eachi∈F. SinceBi=Ai, for eachi∈F, then∪i∈FAicoversXbut a finite amount.
Hence,Xis nearly compact. The proof of the second part of the theorem is similar to the first one and hence omitted.
On the other hand (cf. [20, page 450]) there exist dense-in-itself, compact and ex- tremally disconnected Hausdorff spaces. Such spaces are resolvable and hence cannot bep-closed.
A filter baseᏲ on a topological space(X,τ)is said to pre-θ-converge to a point x∈Xif for eachV∈PO(X,x), there existsF∈Ᏺsuch thatF⊆pcl(V ). A filter base Ᏺis said topre-θ-accumulateatx∈Xif pcl(V )∩F= ∅for everyV∈PO(X,x)and everyF∈Ᏺ. Thepreinteriorof a setA, denoted by pint(A), is the union of all preopen subsets ofA.
Theorem2.8. For a topological space(X,τ)the following conditions are equivalent:
(a) (X,τ)isp-closed,
(b) every maximal filter base pre-θ-converges to some point ofX, (c) every filter base pre-θ-accumulates at some point ofX,
(d) for every family{Vα:α∈A}of preclosed subsets such that∩{Vα:α∈A} = ∅, there exists a finite subsetA0ofAsuch that∩{pint(Vα):α∈A0} = ∅.
Proof. (a)⇒(b). LetᏲbe a maximal filter base onX. Suppose thatᏲdoes not pre- θ-converge to any point ofX. SinceᏲis maximal,Ᏺdoes not pre-θ-accumulate at any point ofX. For eachx∈X, there existFx∈ᏲandVx∈PO(X,x)such that pcl(Vx)∩
Fx= ∅. The family{Vx:x∈X}is a cover ofXby preopen sets ofX. By (a), there exists a finite number of pointsx1,x2,...,xnofXsuch thatX= ∪{pcl(Vxi):i=1,2,...,n}.
SinceᏲis a filter base onX, there existsF0∈Ᏺsuch thatF0⊆ ∩{Fxi:i=1,2,...,n}.
Therefore, we obtainF0= ∅. This is a contradiction.
(b)⇒(c). LetᏲbe any filter base onX. Then, there exists a maximal filter baseᏲ0such thatᏲ⊆Ᏺ0. By (b),Ᏺ0pre-θ-converges to some pointx∈X. For everyF∈Ᏺand every V∈PO(X,x), there existsF0∈Ᏺ0such thatF0⊆pcl(V ); hence∅ =F0∩F⊆pcl(V )∩F. This shows thatᏲpre-θ-accumulates atx.
(c)⇒(d). Let {Vα : α ∈ A} be any family of preclosed subsets of X such that
∩{Vα:α∈A} = ∅. LetΓ(A)denote the ideal of all finite subsets ofA. Assume that
∩{pint(Vα):α∈γ} = ∅for everyγ ∈Γ(A). Then, the familyᏲ= {∩α∈γpint(Vα): γ ∈Γ(A)}is a filter base on X. By (c),Ᏺ pre-θ-accumulates at some point x∈X.
Since{X\Vα:α∈A}is a cover ofX,x∈X\Vα0 for someα0∈A. Therefore, we obtainX\Vα0∈PO(X,x), pint(Vα0)∈Ᏺand pcl(X\Vα0)∩pint(Vα0)= ∅. This is a contradiction.
(d)⇒(a). Let{Vα:α∈A}be a cover ofXby preopen sets ofX. Then{X\Vα:α∈A}is a family of preclosed subsets ofXsuch that∩{X\Vα:α∈A} = ∅. By (d), there exists a finite subsetA0ofAsuch that∩{pint(X\Vα):α∈A0} = ∅; henceX= ∪{pcl(Vα): α∈A0}. This shows thatXisp-closed.
Definition2.9. A topological space(X,τ)is said to bestronglyp-regular(respec- tively,p-regular[8],almostp-regular[14]) if for each pointx∈Xand each preclosed set (respectively, closed set, regular closed set)Fsuch thatx∈F, there exist disjoint preopen setsUandVsuch thatx∈UandF⊆V.
Theorem2.10. If a topological spaceXisp-closed and stronglyp-regular (respec- tively,p-regular, almostp-regular), thenXis strongly compact (respectively, compact, nearly compact).
Proof. We prove only the case ofp-regular spaces. LetX be ap-closed andp- regular space. Let{Vα:α∈A}be any open cover ofX. For eachx∈X, there exists an α(x)∈A such that x∈Vα(x). SinceX is p-regular, there exists U(x)∈PO(X) such thatx∈U(x)⊆pcl(U(x))⊆Vα(x) [8, Theorem 3.2]. Then, {U(x):x∈X}is a preopen cover of thep-closed spaceXand hence there exists a finite amount of points, say,x1,x2,...,xnsuch thatX= ∪ni=1pcl(U(xi))= ∪ni=1Vα(xi). This shows that Xis compact.
3. p-closed subspaces. Recall that a topological space(X,τ)is called hypercon- nectedif every open subset ofXis dense. In the opposite case,Xis calledhyperdis- connected. A setAis calledsemi-regular[5] if it is both semi-open and semi-closed.
Di Maio and Noiri [5] have shown that a setAis semi-regular if and only if there exists a regular open setUwithU⊆A⊆cl(U). Cameron [3] used the term regular semi-open for a semi-regular set.
Lemma3.1(Mashhour et al. [17]). Let A and B be subsets of a topological space (X,τ).
(1)IfA∈PO(X)andB∈SO(X), thenA∩B∈PO(B).
(2)IfA∈PO(B)andB∈PO(X), thenA∈PO(X).
Lemma3.2. LetB⊆A⊆XandA∈SO(X). Then,pclA(B)⊆pclX(B).
Theorem3.3. If every proper semi-regular subspace of a hyperdisconnected topo- logical space(X,τ)isp-closed, thenXis alsop-closed.
Proof. Since(X,τ)is not hyperconnected, then there exists a proper semi-regular setA. Let{Ai}i∈Ibe any preopen cover ofX. SinceAis semi-open, then by Lemma 3.1 Ai∩A=Bi∈PO(A,τ|A). Then{Bi}i∈Iis a preopen cover of thep-closed space(A,τ| A). Then, there exists a finite subsetFofIsuch thatA= ∪i∈FpclA(Bi)⊆ ∪i∈Fpcl(Bi) (by Lemma 3.2). Therefore, we haveA⊆ ∪i∈Fpcl(Ai). SinceAis semi-regular,X\Ais
also semi-regular and by a similar argument we can find a finite subsetGofIsuch that X\A⊆ ∪i∈Gpcl(Ai). Hence,X= ∪i∈F∪Gpcl(Ai). This shows thatXisp-closed.
Theorem3.4. If there exists a proper semi-regular subsetAof a topological space (X,τ)such thatAandX\Aarep-closed subspaces, thenXis alsop-closed.
Proof. The proof is similar to the one of Theorem 3.3 and hence omitted.
Lemma3.5. LetA⊆B⊆XandB∈PO(X). I fA∈PO(B), thenpcl(A)⊆pclB(A).
Theorem3.6. If(X,τ)is ap-closed spaces andAis preregular (i.e., both preopen and preclosed), then(A,τ|A)is alsop-closed (as a subspace).
Proof. Let{Ai}i∈I be any preopen cover of(A,τ|A). By Lemma 3.1,Ai∈PO(X) for eachi∈Iand{Ai:i∈I} ∪(X\A)=X. SinceXisp-closed, there exists a finite subsetFofIsuch thatX=(∪i∈FpclX(Ai))∪(X\A); henceA⊂ ∪i∈FpclX(Ai). For each i∈F, we have by Lemma 3.5, pclX(Ai)⊆pclA(Ai)andA= ∪i∈FpclA(Ai). Therefore, Ais ap-closed subspace.
Example3.7. An open, even aδ-open subset of ap-closed space need not bep- closed (as a subspace). Consider any infinite setXwith the point excluded topology.
Since the only preopen set containing the excluded point is the whole spaceX, then the space in question isp-closed. However, the (infinite) set of isolated points ofXis notp-closed.
4. Sets which arep-closed relative to a space. A subsetSof a topological space (X,τ)is said to bep-closed relative toXif for every cover{Vα:α∈A}ofSby preopen subsets of(X,τ), there exists a finite subsetA0ofAsuch thatS⊂ ∪{pcl(Vα):α∈A0}.
Theorem4.1. For a topological space(X,τ)the following conditions are equivalent:
(a) Sisp-closed relative toX,
(b) every maximal filter base onX which meetsS pre-θ-converges to some point ofS,
(c) every filter base onXwhich meetsSpre-θ-accumulates at some point ofS, (d) for every family{Vα:α∈A}of preclosed subsets of(X,τ)such that [∩{Vα:
α∈A}]∩S= ∅, there exists a finite subsetA0ofAsuch that[∩{pint(Vα):α∈ A0}]∩S= ∅.
A point x∈X is said to be a pre-θ-accumulation point of a subset Aof a topo- logical space(X,τ) if pcl(U)∩A= ∅ for everyU∈PO(X,x). The set of all pre-θ- accumulation points ofAis called thepre-θ-closureofAand is denoted by pclθ(A).
A subsetAof a topological space(X,τ)is said to bepre-θ-closedif pclθ(A)=A. The complement of a pre-θ-closed set is calledpre-θ-open.
Proposition4.2. LetAbe a subsetAof a topological space(X,τ). Then:
(i) IfA∈PO(X), thenpcl(A)=pclθ(A).
(ii) IfAis preregular, thenAis pre-θ-closed.
(iii) IfA∈SO(X), thenpcl(A)=cl(A).
Theorem4.3. IfXis ap-closed space, then every pre-θ-open cover ofXhas a finite subcover.
Proof. Let{Vα:α∈A}be any cover ofXby pre-θ-open subsets ofX. For each x∈X, there existsα(x)∈Asuch that x∈Vα(x). Since Vα(x) is pre-θ-open, there existsVx∈PO(X)such thatx∈Vx⊆pcl(Vx)⊆Vα(x). The family{Vx:x∈X}is a preopen cover ofX. SinceXisp-closed, there exists a finite number of points, say, x1,x2,...,xninXsuch thatX= ∪{pcl(Vxi):i=1,2,...,n}. Therefore, we obtain that X= ∪{Vα(xi):i=1,2,...,n}.
Question4.4. Is the converse in Theorem 4.3 true?
Theorem4.5. LetA, Bbe subsets of a spaceX. I fAis pre-θ-closed andBisp-closed relative toX, thenA∩Bisp-closed relative toX.
Proof. Let{Vα:α∈A}be any cover ofA∩Bby preopen subsets ofX. SinceX\A is pre-θ-open, for eachx∈B\Athere existsWx∈PO(X,x)such that pcl(Wx)⊆X\A.
The family{Wx:x∈B\A}∪{Vα:α∈A}is a cover ofBby preopen sets ofX. Since Bisp-closed relative toX, there exist a finite number of points, say,x1,x2,...,xnin B\Aand a finite subsetA0ofAsuch that
B⊆
∪ni=1pcl Wxi
∪
∪α∈A0pcl Vα
. (4.1)
Since pcl(Wxi)∩A= ∅for eachi, we obtain thatA∩B⊆ ∪{pcl(Vα):α∈A0}. This shows thatA∩Bisp-closed relative toX.
Corollary4.6. IfKis pre-θ-closed set of ap-closed space(X,τ), thenKisp-closed relative toX.
Question4.7. If in a topological space(X,τ)every proper pre-θ-closed set isp- closed relative toX, isXnecessarilyp-closed?
A topological space(X,τ)is calledpreconnected[19] ifXcannot be expressed as the union of two disjoint preopen sets. In the opposite case,Xis calledpredisconnected.
Note that every preconnected space is irresolvable but not vice versa.
Theorem4.8. LetXbe a predisconnected space. ThenXisp-closed if and only if every preregular subset ofXisp-closed relative toX.
Proof
Necessity. Every preregular set is pre-θ-closed by Proposition 4.2. SinceXis p- closed, the proof is completed by Corollary 4.6.
Sufficiency. Let{Vα:α∈A}be a preopen cover ofX. SinceXis predisconnected, there exists a proper preregular subsetAofX. By our hypothesis,AandX\Aarep- closed relative toX. There exist finite subsetsA1andA2ofAsuch that
A⊆ ∪α∈A1pcl Vα
, X\A⊆ ∪α∈A2pcl Vα
. (4.2)
Therefore, we obtain thatX= ∪{pcl(Vα):α∈A1∪A2}.
Theorem4.9. If there exists a proper preregular subsetAof a topological space (X,τ)such thatAandX\Aarep-closed relative toX, thenXisp-closed.
Proof. This proof is similar to the one of Theorem 4.8 and hence omitted.
Theorem4.10. LetX0be a semi-open subset of a topological space(X,τ). I fX0is ap-closed space, then it isp-closed relative toX.
Proof. Let{Vα:α∈A}be any cover ofX0by preopen subsets ofX. SinceX0∈ SO(X), by Lemma 3.1, we have thatX0∩Vα=Wα∈PO(X0)for eachα∈A. Therefore, {Wα:α∈A}is a preopen cover ofX0. sinceX0isp-closed, there exists a finite subset A0 ofAsuch thatX0= ∪{pclX0(Wα):α∈A0}. By Lemma 3.2, we obtain thatX0⊆
∪{pcl(Wα):α∈A0} ⊆ ∪{pcl(Vα):α∈A0}. This shows thatX0isp-closed relative toX.
Theorem4.11. LetX0be a preopen subset of a topological space(X,τ). I fX0is a p-closed relative toX, then it is ap-closed subspace ofX.
Proof. Let{Vα:α∈A}be any cover ofX0by preopen subsets ofX0. SinceX0∈ PO(X), by Lemma 3.1,Vα∈PO(X)for eachα∈A. SinceX0 is p-closed relative to X, there exists a finite subsetA0 of Asuch that X0⊆ ∪ {pcl(Vα):α∈A0}. Since X0∈PO(X), by Lemma 3.5 we obtainX0= ∪{pclX0(Vα):α∈A0}.This shows thatX0
is ap-closed subspace ofX.
Corollary4.12. LetX0be an (α-)open subset of a topological space(X,τ). Then X0is ap-closed subspace ofXif and only if it isp-closed relative toX.
Proof. This is an immediate consequence of Theorems 4.10 and 4.11.
Recall that a functionf :(X,τ)→(Y ,σ ) is calledpreirresolute[23] (respectively, precontinuous[15]) iff−1(V)is preopen inXfor every preopen (respectively, open) subsetVofY.
Lemma4.13(see [18]). A functionf:(X,τ)→(Y ,σ )is preirresolute (respectively, precontinuous) if and only if for each subsetAofX, f (pcl(A))⊆pcl(f (A))(respec- tively,f (pcl(A))⊆cl(f (A))).
Theorem4.14. If a functionf:(X,τ)→(Y ,σ )is a preirresolute (respectively, pre- continuous) surjection andKis p-closed relative toX, thenf (K)isp-closed (respec- tively, QHC) relative toY.
Proof. Let{Vα:α∈A}be any cover off (K)by preopen (respectively, open) sub- sets of Y. Since f is preirresolute (respectively, precontinuous),{f−1(Vα):α∈A}
is a cover ofK by preopen subsets ofX, where Kis p-closed relative toX. There- fore, there exists a finite subsetA0ofAsuch thatK⊆ ∪α∈A0pcl(f−1(Vα)). Sincefis preirresolute (respectively, precontinuous) and surjective, by Lemma 4.13, we have
f (K)⊆∪α∈A0f pcl
f−1 Vα
⊆ ∪α∈A0pcl Vα respectively,f (K)⊆ ∪α∈A0f
pcl f−1
Vα
⊆ ∪α∈A0cl Vα
. (4.3) Corollary4.15. If a functionf :(X,τ)→(Y ,σ ) is a preirresolute (respectively, continuous) surjection andXisp-closed, thenY isp-closed (respectively, QHC).
Corollary4.16. (i)The property “p-closed” is topological.
(ii) If the product space
α∈AXαisp-closed, thenXαisp-closed for eachα∈A.
Remark4.17. Even finite product ofp-closed spaces need not be p-closed; for consider the product of the space from Example 2.6(i) with any two point indiscrete space. This product space shows that [1, Theorem 3.4.3] is wrong, i.e., every proper preregular subset might bep-closed relative to the space and still the space might fail to bep-closed. Additionally, [1, Example 3.4.1] is also false.
Acknowledgement. Research is supported partially by the Ella and Georg Ehrnooth Foundation at Merita Bank, Finland and by the Japan-Scandinavia Sasakawa Foundation.
References
[1] Abd El-Aziz Abo-Khadra,On generalized forms of compactness, Master’s thesis, Faculty of Science, Tanta University, Egypt, 1989.
[2] A. V. Arhangel’ski˘i and P. J. Collins,On submaximal spaces, Topology Appl.64(1995), no. 3, 219–241. MR 96m:54002. Zbl 826.54002.
[3] D. E. Cameron,Properties ofS-closed spaces, Proc. Amer. Math. Soc. 72(1978), no. 3, 581–586. MR 58#24188. Zbl 408.54018.
[4] H. H. Corson and E. Michael,Metrizability of certain countable unions, Illinois J. Math.8 (1964), 351–360. MR 30#562. Zbl 127.13203.
[5] G. Di Maio and T. Noiri,Ons-closed spaces, Indian J. Pure Appl. Math.18(1987), no. 3, 226–233. MR 88d:54029. Zbl 625.54031.
[6] J. Dontchev and M. Ganster,On covering spaces with semi-regular sets, Ricerche Mat.45 (1996), no. 1, 229–245. MR 98d:54040. Zbl 990.35704.
[7] J. Dontchev, M. Ganster, and D. Rose,α-scattered spaces II, Houston J. Math.23(1997), no. 2, 231–246. Zbl 894.54033.
[8] N. El-Deeb, I. A. Hasanein, A. S. Mashhour, and T. Noiri,Onp-regular spaces, Bull. Math.
Soc. Sci. Math. R. S. Roumanie (N.S.)27(75)(1983), no. 4, 311–315. MR 85d:54018.
Zbl 524.54016.
[9] J. Foran and P. Liebnitz,A characterization of almost resolvable spaces, Rend. Circ. Mat.
Palermo (2)40(1991), no. 1, 136–141. MR 92g:54044. Zbl 725.54024.
[10] M. Ganster,Preopen sets and resolvable spaces, Kyungpook Math. J.27 (1987), no. 2, 135–143. MR 90e:54004. Zbl 665.54001.
[11] , Some remarks on strongly compact spaces and semi-compact spaces, Bull.
Malaysian Math. Soc. (2)10(1987), no. 2, 67–70. MR 90e:54052. Zbl 668.54016.
[12] D. S. Jankovi´c, I. L. Reilly, and M. K. Vamanamurthy,On strongly compact topological spaces, Questions Answers Gen. Topology6(1988), no. 1, 29–40. MR 89c:54052.
Zbl 647.54018.
[13] E. H. Kronheimer,The topology of digital images, Topology Appl.46(1992), no. 3, 279–
303. MR 94a:54089. Zbl 770.54035.
[14] S. R. Malghan and G. B. Navalagi,Almostp-regular,p-completely regular and almostp- completely regular spaces, Bull. Math. Soc. Sci. Math. Roumanie (N.S.)34(82)(1990), no. 4, 317–326. CMP 1 309 531. Zbl 752.54005.
[15] A. S. Mashhour, M. E. Abd El-Monsef, and S. N. El-Deep,On precontinuous and weak pre- continuous mappings, Proc. Math. Phys. Soc. Egypt53(1982), 47–53. MR 87c:54002.
Zbl 571.54011.
[16] A. S. Mashhour, M. E. Abd El-Monsef, I. A. Hasanein, and T. Noiri,Strongly compact spaces, Delta J. Sci.8(1984), 30–46.
[17] A. S. Mashhour, I. A. Hasanein, and S. N. El-Deeb,A note on semicontinuity and precon- tinuity, Indian J. Pure Appl. Math.13(1982), no. 10, 1119–1123. MR 84a:54021.
Zbl 499.54009.
[18] V. Popa,Characterizations of H-almost continuous functions, Glas. Mat. Ser. III22(42) (1987), no. 1, 157–161. MR 89c:54028. Zbl 644.54009.
[19] , Properties ofH-almost continuous functions, Bull. Math. Soc. Sci. Math. R. S.
Roumanie (N.S.)31(79)(1987), no. 2, 163–168. MR 88k:54028. Zbl 618.54013.
[20] J. R. Porter and R. G. Woods,Extensions and Absolutes of Hausdorff Spaces, Springer- Verlag, New York, 1988. MR 89b:54003. Zbl 652.54016.
[21] S. Raychaudhuri and M. N. Mukherjee,Onδ-almost continuity andδ-preopen sets, Bull.
Inst. Math. Acad. Sinica21(1993), no. 4, 357–366. CMP 1 256 986. Zbl 808.54010.
[22] ,δp-closedness for topological spaces, J. Indian Acad. Math.18(1996), no. 1, 89–99.
CMP 1 469 887. Zbl 893.54019.
[23] I. L. Reilly and M. K. Vamanamurthy,Onα-continuity in topological spaces, Acta Math.
Hungar.45(1985), no. 1-2, 27–32. MR 86h:54013. Zbl 576.54014.
[24] M. K. Singal and A. Mathur,On nearly-compact spaces, Boll. Un. Mat. Ital. (4)2(1969), 702–710. MR 41#2628. Zbl 188.28005.
[25] N. V. Veliˇcko, H-closed topological spaces, Mat. Sb. (N.S.) 70(112) (1966), 98–112.
MR 33#6576. Zbl 183.27302.
Julian Dontchev: Department of Mathematics, University of Helsinki, PL4, Yliopis- tonkatu15,00014Helsinki, Finland
E-mail address:[email protected], [email protected]
Maximilian Ganster: Department of Mathematics, Graz University of Technology, Steyrergasse30, A-8010Graz, Austria
E-mail address:[email protected]
Takashi Noiri: Department of Mathematics, Yatsushiro College of Technology,2627 Hirayama Shinmachi, Yatsushiro-shi, Kumamoto-ken,866-8501, Japan
E-mail address:[email protected]
