PII. S0161171201006500 http://ijmms.hindawi.com © Hindawi Publishing Corp.
ON
θ-PRECONTINUOUS FUNCTIONS
TAKASHI NOIRI
(Received 16 January 2001)
Abstract.We introduce a new class of functions calledθ-precontinuous functions which is contained in the class of weakly precontinuous (or almost weakly continuous) func-tions and contains the class of almost precontinuous funcfunc-tions. It is shown that theθ -precontinuous image of ap-closed space is quasiH-closed.
2000 Mathematics Subject Classification. 54C08.
1. Introduction. A subsetAof a topological spaceXis said to be preopen [14] or nearly open [26] ifA⊂Int(Cl(A)). A functionf:X→Y is called precontinuous [14] if the preimagef−1(V )of each open setV ofY is preopen inX. Precontinuity was called near continuity by Pták [26] and also called almost continuity by Frolík [9] and Husain [10]. In 1985, Jankovi´c [12] introduced almost weak continuity as a weak form of precontinuity. Popa and Noiri [23] introduced weak precontinuity and showed that almost weak continuity is equivalent to weak precontinuity. Paul and Bhattacharyya [21] called weakly precontinuous functions quasi precontinuous and obtained the fur-ther properties of quasi precontinuity. Recently, Nasef and Noiri [16] have introduced and investigated the notion of almost precontinuity. Quite recently, Jafari and Noiri [11] investigated the further properties of almost precontinuous functions.
In this paper, we introduce a new class of functions calledθ-precontinuous func-tions which is contained in the class of weakly precontinuous funcfunc-tions and con-tains the class of almost precontinuous functions. We obtain basic properties ofθ -precontinuous functions. It is shown in the last section that the θ-precontinuous images of p-closed (resp., β-connected) spaces are quasi H-closed (resp., semi-connected).
2. Preliminaries. Throughout, by(X, τ)and(Y , σ )(or simplyXandY) we denote topological spaces. LetSbe a subset ofX. We denote the interior and the closure of
Sby Int(S)and Cl(S), respectively. A subsetS is said to bepreopen[14] (resp., semi-open[13],α-open[17]) ifS⊂Int(Cl(S))(resp.,S⊂Cl(Int(S)),S⊂Int(Cl(Int(S)))). The complement of a preopen set is calledpreclosed. The intersection of all preclosed sets containingSis called thepreclosure[8] ofSand is denoted by pCl(S). Thepreinterior
of S is defined by the union of all preopen sets contained in S and is denoted by pInt(S). The family of all preopen sets ofXis denoted by PO(X). We set PO(X, x)= {U:
is called a preθ-cluster point ofS if pCl(U )∩S = ∅for every preopen setU ofX
containingx. The set of all preθ-cluster points ofS is called thepreθ-closureofS
and is denoted by pClθ(S). A subsetS is said to bepreθ-closed[20] ifS=pClθ(S). The complement of a preθ-closed set is said to bepreθ-open.
Definition2.1. A functionf:X→Yis said to beprecontinuous[14] (resp.,almost
precontinuous[16],weakly precontinuous[23] orquasi precontinuous[21]) if for each
x∈Xand each open setVofY containingf (x), there existsU∈PO(X, x)such that
f (U )⊂V (resp.,f (U )⊂Int(Cl(V )),f (U )⊂Cl(V )).
Definition2.2. A functionf:X→Y is said to bealmost weakly continuous [12]
iff−1(V )⊂Int(Cl(f−1(Cl(V ))))for every open setV ofY.
Definition2.3. A functionf:X→Y is said to bestronglyθ-precontinuous[19] if for eachx∈Xand each open setV ofY containingf (x), there existsU∈PO(X, x)
such thatf (pCl(U ))⊂V.
Definition2.4. A function f :X→Y is said to be θ-precontinuous if for each
x∈Xand each open setVofY containingf (x), there existsU∈PO(X, x)such that
f (pCl(U ))⊂Cl(V ).
Remark2.5. By the above definitions andTheorem 3.3below, we have the following
implications and none of these implications is reversible by [19, Example 2.2], [11, Example 2.9], and Examples2.6and5.11below.
stronglyθ-precontinuous ⇒precontinuous ⇒almost precontinuous
⇒θ-precontinuous ⇒weakly precontinuous. (2.1)
Example 2.6. This example is due to Arya and Deb [4]. Let X be the set of all
real numbers. The topology τ onX is the cocountable topology. LetY = {a, b, c},
σ = {∅, Y ,{a},{c},{a, c}}. We define a functionf:(X, τ)→(Y , σ )byf (x)=aifx
is rational;f (x)=bifxis irrational. Thenfis aθ-precontinuous function which is not almost precontinuous
3. Characterizations
Theorem3.1. For a functionf:X→Y the following properties are equivalent: (1) f isθ-precontinuous;
(2) pClθ(f−1(B))⊂f−1(Clθ(B))for every subset B of Y; (3) f (pClθ(A))⊂Clθ(f (A))for every subset A of X.
Proof. (1)⇒(2). Let B be any subset of Y. Suppose that x ∉f−1(Clθ(B)). Then f (x)∉Clθ(B)and there exists an open setVcontainingf (x)such that Cl(V )∩B= ∅. Sincefisθ.p.c., there existsU∈PO(X, x)such thatf (pCl(U ))⊂Cl(V ). Therefore, we havef (pCl(U ))∩B= ∅and pCl(U )∩f−1(B)= ∅. This shows thatx∉pCl
(2)⇒(3). Let A be any subset of X. Then we have pClθ(A)⊂pClθ(f−1(f (A)))⊂ f−1(Cl
θ(f (A)))and hencef (pClθ(A))⊂Clθ(f (A)).
(3)⇒(2). LetBbe a subset ofY. We havef (pClθ(f−1(B)))⊂Clθ(f (f−1(B)))⊂Clθ(B) and hence pClθ(f−1(B))⊂f−1(Clθ(B)).
(2)⇒(1). Letx∈XandVbe an open set ofY containingf (x). Then we have Cl(V )∩ (Y−Cl(V ))= ∅andf (x)∉Clθ(Y−Cl(V )). Hence,x∉f−1(Clθ(Y−Cl(V )))andx∉ pClθ(f−1(Y−Cl(V ))). There existsU∈PO(X, x)such that pCl(U )∩f−1(Y−Cl(V ))= ∅; hencef (pCl(U ))⊂Cl(V ). Therefore,fisθ.p.c.
Theorem3.2. For a functionf:X→Y the following properties are equivalent: (1) fisθ-precontinuous;
(2) f−1(V )⊂pInt
θ(f−1(Cl(V )))for every open setVofY; (3) pClθ(f−1(V ))⊂f−1(Cl(V ))for every open setVofY.
Proof. (1)⇒(2). Suppose thatVis any open set ofY andx∈f−1(V ). Thenf (x)∈ V and there existsU∈PO(X, x)such that f (pCl(U ))⊂Cl(V ). Therefore,x ∈U⊂
pCl(U )⊂f−1(Cl(V )). This shows thatx∈pInt
θ(f−1(Cl(V ))). Therefore, we obtain f−1(V )⊂pInt
θ(f−1(Cl(V ))).
(2)⇒(3). Suppose thatV is any open set ofY and x ∉f−1(Cl(V )). Thenf (x)∉ Cl(V )and there exists an open setW containing f (x)such thatW∩V = ∅; hence Cl(W )∩V= ∅. Therefore, we havef−1(Cl(W ))∩f−1(V )= ∅. Sincex∈f−1(W ), by (2)x∈pIntθ(f−1(Cl(W ))). There existsU∈PO(X, x)such that pCl(U )⊂f−1(Cl(W )). Thus we have pCl(U )∩f−1(V )= ∅ and hencex ∉pCl
θ(f−1(V )). This shows that pClθ(f−1(V ))⊂f−1(Cl(V )).
(3)⇒(1). Suppose that x∈X and V is any open set of Y containingf (x). Then
V∩(Y−Cl(V ))= ∅andf (x)∉Cl(Y−Cl(V )). Therefore,x∉f−1(Cl(Y−Cl(V )))and by (3)x∉pClθ(f−1(Y−Cl(V ))). There existsU∈PO(X, x)such that pCl(U )∩f−1(Y− Cl(V ))= ∅. Therefore, we obtainf (pCl(U ))⊂Cl(V ). This shows thatfisθ.p.c.
Theorem3.3. For a functionf:X→Y the following properties hold:
(1) iffis almost precontinuous, then it isθ-precontinuous;
(2) iffisθ-precontinuous, then it is weakly precontinuous.
Proof. Statement (2) is obvious. We will show statement (1). Suppose thatx∈ X and V is any open set of Y containing f (x). Since f is almost precontinuous,
f−1(Int(Cl(V )))is preopen andf−1(Cl(V ))is preclosed inXby [16, Theorem 3.1]. Now, setU=f−1(Int(Cl(V ))). Then we haveU∈PO(X, x)and pCl(U )⊂f−1(Cl(V )). Therefore, we obtainf (pCl(U ))⊂Cl(V ). This shows thatfisθ.p.c.
Corollary3.4. Let Y be a regular space. Then, for a functionf:X→Ythe following properties are equivalent:
(1) fis stronglyθ-precontinuous;
(2) fis precontinuous;
(3) fis almost precontinuous;
(4) fisθ-precontinuous;
(5) fis weakly precontinuous.
Definition3.5. A topological space Xis said to be pre-regular [20] if for each
preclosed setF and each pointx∈X−F, there exist disjoint preopen setsUandV
such thatx∈UandF⊂V.
Lemma3.6(see [20]). A topological spaceXis pre-regular if and only if for eachU∈ PO(X)and each pointx∈U, there existsV∈PO(X, x)such thatx∈V⊂pCl(V )⊂U.
Theorem3.7. Let X be a pre-regular space. Thenf:X→Y isθ.p.c.if and only if it
is weakly precontinuous.
Proof. Suppose thatf is weakly precontinuous. Letx∈XandVis any open set ofYcontainingf (x). Then, there existsU∈PO(X, x)such thatf (U )⊂Cl(V ). SinceX
is pre-regular, there existsU∗∈PO(X, x)such thatx∈U∗⊂pCl(U∗)⊂U. Therefore,
we obtainf (pCl(U∗))⊂Cl(V ). This shows thatf isθ.p.c.
Theorem3.8. Letf:X→Y be a function andg:X→X×Y the graph function of
fdefined byg(x)=(x, f (x))for eachx∈X. Then g isθ.p.c.if and only iffisθ.p.c.
Proof
Necessity.Suppose thatgisθ.p.c.Letx∈XandVbe an open set ofYcontaining f (x). ThenX×Vis an open set ofX×Ycontainingg(x). Sincegisθ.p.c., there exists
U∈PO(X, x)such thatg(pCl(U ))⊂Cl(X×V ). It follows that Cl(X×V )=X×Cl(V ). Therefore, we obtainf (pCl(U ))⊂Cl(V ). This shows thatfisθ.p.c.
Sufficiency. Letx∈X andW be any open set ofX×Y containingg(x). There exist open setsU1⊂XandV⊂Y such thatg(x)=(x, f (x))∈U1×V⊂W. Sincef isθ.p.c., there existsU2∈PO(X, x)such thatf (pCl(U2))⊂Cl(V ). LetU =U1∩U2, thenU∈PO(X, x). Therefore, we obtaing(pCl(U ))⊂Cl(U1)×f (pCl(U2))⊂Cl(U1)× Cl(V )⊂Cl(W ). This shows thatgisθ.p.c.
4. Some properties
Lemma4.1(see [15]). LetAandX0be subsets of a spaceX. (1)IfA∈PO(X)andX0is semi-open inX, thenA∩X0∈PO(X0). (2)IfA∈PO(X0)andX0∈PO(X), thenA∈PO(X).
Lemma4.2(see [7]). LetAandX0be subsets of a spaceXsuch thatA⊂X0⊂X. Let pClX0(A)denote the preclosure ofAin the subspaceX0.
(1)IfX0is semi-open inX, thenpClX0(A)⊂pCl(A).
(2)IfA∈PO(X0)andX0∈PO(X), thenpCl(A)⊂pClX0(A).
Theorem 4.3. Iff :X→Y is θ.p.c.and X0 is a semi-open subset ofX, then the restrictionf /X0:X0→Y isθ.p.c.
Proof. For anyx∈X0and any open neighborhoodV off (x), there existsU∈ PO(X, x) such thatf (pCl(U ))⊂Cl(V ) since f is θ.p.c. Put U0=U∩X0, then by Lemmas4.1and4.2U0∈PO(X0, x)and pClX0(U0)⊂pCl(U0). Therefore, we obtain
f /X0
pClX0
U0
=fpClX0
U0
⊂fpClU0
⊂fpCl(U )⊂Cl(V ). (4.1)
Theorem4.4. A functionf :X→Y isθ.p.c. if for eachx∈X there existsX0∈ PO(X, x)such that the restrictionf /X0:X0→Y isθ.p.c.
Proof. Letx∈X and V be any open neighborhood off (x). There existsX0∈ PO(X, x) such that f /X0:X0→Y is θ.p.c. Thus, there existsU ∈PO(X0, x) such that(f /X0)(pClX0(U ))⊂Cl(V ). By Lemmas4.1and4.2,U∈PO(X, x)and pCl(U )⊂
pClX0(U ). Hence, we havef (pCl(U ))=(f /X0)(pCl(U ))⊂(f /X0)(pClX0(U ))⊂Cl(V ).
This shows thatfisθ.p.c.
Corollary4.5. Let{Uλ:λ∈Λ}be an α-open cover of a topological spaceX. A functionf:X→Yisθ.p.c.if and only if the restrictionf /Uλ:Uλ→Y isθ.p.c.for each λ∈Λ.
Proof. This is an immediate consequence of Theorems4.3and4.4.
Let{Xα:α∈Ꮽ}be a family of topological spaces,Aαa nonempty subset ofXαfor eachα∈ᏭandX=Π{Xα:α∈Ꮽ}denote the product space, whereᏭis nonempty.
Lemma4.6(see [8]). Letnbe a positive integer andA=Πnj=1Aα
j×Πα=αjXα. (1)A∈PO(X)if and only ifAαj∈PO(Xαj)for eachj=1,2, . . . , n.
(2) pCl(Πα∈ᏭAα)⊂Πα∈ᏭpCl(Aα).
Theorem4.7. If a functionfα:Xα→Yαisθ.p.c.for eachα∈Ꮽ. Then the product functionf:ΠXα→ΠYα, defined byf ({xα})= {fα(xα)}for eachx= {xα}, isθ.p.c.
Proof. Letx= {xα} ∈ΠXαandWbe any open set ofΠYαcontainingf (x). Then, there exists an open setVαjofYαj such that
f (x)=fα
xα
∈Πn
j=1Vαj×Πα=αjYα⊂W . (4.2) Sincefαisθ.p.c.for eachα, there existsUαj∈PO(Xαj, xαj)such thatfαj(pCl(Uαj)) ⊂Cl(Vαj)forj=1,2, . . . , n. Now, putU=Π
n
j=1Uαj×Πα=αjXα. Then, it follows from
Lemma 4.6thatU∈PO(ΠXα, x). Moreover, we have
fpCl(U )⊂fΠnj=1pClUαj
×Πα=αjXα
⊂Πnj=1fαj
pClUαj
×Πα=αjYα ⊂Πn
j=1Cl
Vαj
×Πα=αjYα⊂Cl(W ).
(4.3)
This shows thatfisθ.p.c.
5. Preservation property
Definition5.1. A topological spaceXis said to be
(1) p-closed [7] (resp.,p-Lindelöf) if every cover ofXby preopen sets has a finite (resp., countable) subfamily whose preclosures coverX,
(2) countably p-closed if every countable cover ofX by preopen sets has a finite subfamily whose preclosures coverX;
(3) quasi H-closed[25] (resp.,almost Lindelöf [6]) if every cover ofXby open sets has a finite (resp., countable) subfamily whose closures coverX,
Definition5.2. A subsetKof a spaceXis said to be
(1) p-closed relative toX[7] if for every cover{Vα:α∈Ꮽ}ofKby preopen sets ofX, there exists a finite subsetᏭ∗ofᏭsuch thatK⊂ ∪{pCl(Vα):α∈Ꮽ∗},
(2) quasi H-closed relativetoX[25] if for every cover{Vα:α∈Ꮽ}ofKby open sets ofX, there exists a finite subsetᏭ∗ofᏭsuch thatK⊂ ∪{Cl(Vα):α∈Ꮽ∗}.
Theorem5.3. Iff:X→Y is aθ.p.c.function and K isp-closed relative toX, then f (K)is quasiH-closed relative toY.
Proof. Suppose thatf:X→Y isθ.p.c.andKisp-closed relative toX. Let{Vα:
α∈Ꮽ}be a cover of f (K)by open sets of Y. For each point x ∈K, there exists
α(x)∈Ꮽsuch thatf (x)∈Vα(x). Sincef isθ.p.c., there existsUx∈PO(X, x)such thatf (pCl(Ux))⊂Cl(Vα(x)). The family{Ux:x∈K}is a cover ofKby preopen sets of X and hence there exists a finite subset K∗ of K such that K⊂ ∪x∈K∗pCl(Ux). Therefore, we obtainf (K)⊂ ∪x∈K∗Cl(Vα(x)). This shows thatf (K)is quasiH-closed relative toY.
Corollary5.4. Letf:X→Y be aθ.p.c.surjection. Then, the following properties hold:
(1)IfXisp-closed, thenY is quasiH-closed.
(2)IfXisp-Lindelöf, thenY is almost Lindelöf.
(3)IfXis countablyp-closed, thenY is lightly compact.
A subset S of a topological spaceX is said to beβ-open [1] or semipreopen[3] ifS⊂Cl(Int(Cl(S))). It is well known thatα-openness implies both preopenness and semi-openness which implyβ-openness. The complement of a semipreopen set is said to besemipreclosed [3]. The intersection of all semipreclosed sets ofXcontaining a subsetSis thesemipreclosureofS and is denoted by spCl(S)[3].
Definition5.5. A topological spaceXis said to be
(1) β-connected[24] orsemipreconnected[2] ifXcannot be expressed as the union of two nonempty disjointβ-open sets,
(2) semi-connected [22] ifX cannot be expressed as the union of two nonempty disjoint semi-open sets.
Remark5.6. We have the following implications:
β-connected ⇒semi-connected ⇒connected. (5.1)
But, the converses need not be true as the following simple examples show.
Example5.7. (1) Let X= {a, b, c}and τ = {X,∅,{a},{b},{a, b}}. Then (X, τ)is connected but not semi-connected.
(2) LetX= {a, b, c}and τ= {X,∅,{b, c}}. Then(X, τ)is semi-connected but not
β-connected.
Lemma5.8. For a topological spaceX, the following properties are equivalent:
(1)Xisβ-connected or semipreconnected.
(2)The intersection of two nonempty semipreopen subsets ofXis always nonempty.
(4) pCl(V )=Xfor every nonempty preopen subsetVof X.
(5) spCl(V )=Xfor every nonempty semipreopen subsetVof X.
Proof. The proofs of equivalences of (1), (2), and (3) are given in [2, Theorem 6.4].
The other properties (4) and (5), which are stated in [18], are easily equivalent to (3) and (2), respectively.
Theorem5.9. Iff :X→Y is a θ.p.c.surjection andX is β-connected, thenY is semi-connected.
Proof. LetV be any nonempty open set ofY. Lety ∈V. Sincef is surjective, there existsx∈Xsuch thatf (x)=y. Sincef isθ.p.c., there existsU∈PO(X, x)
such thatf (pCl(U ))⊂Cl(V ). SinceXisβ-connected, byLemma 5.8pCl(U )=Xand hence Cl(V )=Y sincef is surjective. Therefore, it follows from [22, Theorem 4.3] thatY is semi-connected.
Remark 5.10. The following example shows that the image ofβ-connectedness
under weakly precontinuous surjections is not necessarily semi-connected.
Example 5.11. Let X be the set of real numbers, τ = {∅} ∪ {V ⊂ X : 0∈ V}, Y = {a, b, c}, andσ = {Y ,∅,{a},{b},{a, b}}. Define a functionf:(X, τ)→(Y , σ )as follows:f (x)=aifx <0;f (x)=cifx=0;f (x)=bifx >0. Thenfis a weakly pre-continuous surjection which is notθ.p.c.The topological space(X, τ)isβ-connected byLemma 5.8. ByExample 5.7(1),(Y , σ )is connected but not semi-connected.
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Takashi Noiri: Department of Mathematics, Yatsushiro College of Technology, Yatsushiro, Kumamoto,866-8501, Japan
