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SLIGHTLY β-CONTINUOUS FUNCTIONS
TAKASHI NOIRI
(Received 1 September 2000 and in revised form 30 January 2001)
Abstract.We define a functionf:X→Yto be slightlyβ-continuous if for every clopen setVofY,f−1(V )⊂Cl(Int(Cl(f−1(V )))). We obtain several properties of such a function.
Especially, we define the notion of ultra-regularizations of a topology and obtain interest- ing characterizations of slightlyβ-continuous functions by using it.
2000 Mathematics Subject Classification. 54C08.
1. Introduction. Semi-open sets, preopen sets,α-sets, andβ-open sets play an im- portant role in the researches of generalizations of continuity in topological spaces.
By using these sets many authors introduced and studied various types of general- izations of continuity. In 1980 Jain [15] introduced the notion of slightly continuous functions. Recently, Nour [24] defined slightly semi-continuous functions as a weak form of slight continuity and investigated the functions. Quite recently, Noiri and Chae [23] have further investigated slightly semi-continuous functions. On the other hand, Pal and Bhattacharyya [7] defined a function to be faintly precontinuous if the preimages of each clopen set of the codomain is preopen and obtained many proper- ties of such functions. Slight continuity implies both slight semi-continuity and faint precontinuity but not conversely.
In this paper, we introduce the notion of slightβ-continuity which is implied by both slight semi-continuity and faint precontinuity. We establish several properties of such functions. Especially, we define the notion of ultra-regularization of a topology and obtain interesting characterizations of slightβ-continuity, slight semi-continuity, faint precontinuity and slight continuity. Moreover, we investigate the relationships between slightβ-continuity, contra-β-continuity [13], andβ-continuity [1].
2. Preliminaries. Let(X, τ)be a topological space andAa subset ofX. The closure ofAand the interior ofAare denoted by Cl(A)and Int(A), respectively. A subsetAis said to beβ-open[1] orsemi-preopen[5] (resp.,semi-open[17],preopen[19],α-open [21]) ifA⊂Cl(Int(Cl(A)))(resp.,A⊂Cl(Int(A)),A⊂Int(Cl(A)),A⊂Int(Cl(Int(A)))).
The family of all semi-open (resp., preopen,α-open,β-open) sets in(X, τ)is denoted by SO(X)(resp., PO(X), α(X), β(X), or SPO(X)). The complement of a semi-open (resp., preopen,α-open,β-open) set is said to besemi-closed(resp.,preclosed,α-closed, β-closed, or semi-preclosed). IfA is both semi-open and semi-closed, then it is said to besemi-regular [9]. IfA is bothβ-open andβ-closed, then it is said to besemi- pre-regularorβ-clopen. The family of all semi-regular (resp., semi-preopen, semi-pre- regular, clopen) sets ofX is denoted by SR(X)(resp., SPO(X), SPR(X), CO(X)). The family of all clopen (resp., semi-preopen, semi-pre-regular) sets ofXcontainingx∈
Xis denoted by CO(X, x)(resp., SPO(X, x), SPR(X, x)). The intersection of all semi- closed (resp., preclosed,β-closed) sets ofX containing Ais called thesemi-closure [8] (resp.,preclosure[11],semi-preclosure[5] orβ-closure[3]) ofAand is denoted by sCl(A)(resp., pCl(A), spCl(A), orβCl(A)).
The following basic properties of the semi-preclosure are useful in the sequel.
Lemma2.1(see Abd El-Monsef et al. [3] and Andrijevi´c [5]). The following statements hold for a subset A of a topological space(X, τ):
(a) spCl(A)=A∪Int(Cl(Int(A))),
(b) x∈spCl(A)if and only ifA∩U= ∅for everyU∈SPO(X, x), (c) A isβ-closed if and only ifA=spCl(A).
Lemma2.2(see Jafari and Noiri [14]). If Ais aβ-open set of a topological space (X, τ), thenspCl(A)isβ-open in(X, τ).
Throughout the present paper,(X, τ)and(Y , σ )(or simplyXandY) denote topo- logical spaces andf:(X, τ)→(Y , σ )(or simplyf:X→Y) presents a (single-valued) function.
Definition2.3. A functionf:(X, τ)→(Y , σ )is said to beslightly continuous[15]
(resp.,slightly semi-continuous [24],faintly precontinuous[7]) if for each pointx∈X and each clopen setVcontainingf (x)there exists an open setU (resp.,U∈SO(X), U∈PO(X)) containingxsuch thatf (U )⊂V.
Definition2.4. A functionf:(X, τ)→(Y , σ )is said to beβ-continuous[1] (resp., semi-continuous[17],precontinuous[19]) if for each pointx∈Xand each open setV containingf (x)there existsU∈SPO(X)(resp.,U∈SO(X),U∈PO(X)) containingx such thatf (U )⊂V.
3. Characterizations
Definition3.1. A functionf:(X, τ)→(Y , σ )is said to beslightlyβ-continuous (brieflysl.β.c.) if for each pointx∈X and each clopen setV containingf (x)there exists aβ-open setUofXcontainingxsuch thatf (U )⊂V.
Theorem3.2. For a functionf:(X, τ)→(Y , σ ), the following statements are equiv- alent:
(a) fis slightlyβ-continuous;
(b) f−1(V )∈SPO(X)for eachV∈CO(Y );
(c) f−1(V )∈SPR(X)for eachV∈CO(Y );
(d) for eachx∈Xand eachV∈CO(Y , f (x)), there existsU∈SPR(X, x)such that f (U )⊂V;
(e) for eachx∈Xand eachV∈CO(Y , f (x)), there existsU∈SPO(X, x)such that f (spCl(U ))⊂V.
Proof. The proof is easily obtained by usingLemma 2.2.
Let(X, τ)be a topological space. Since the intersection of two clopen sets of(X, τ) is clopen, the clopen subsets of(X, τ)may be used as a base for a topology onX. The
topology is called theultra-regularization ofτ and is denoted byτu. A topological space(X, τ)is said to beultra regular [12] ifτ=τu. Each element ofτuis said to be δ∗-open[29]. Note that ultra-regular spaces are known as 0-dimensional spaces.
Definition3.3. A functionf:(X, τ)→(Y , σ )is said to beclopen-continuous[28]
if for each pointxofXand each open setV containingf (x), there exists a clopen setUcontainingxsuch thatf (U )⊂V.
Remark3.4. A space(X, τ)is ultra-regular if and only if every continuous function f:(X, τ)→(Y , σ )is clopen-continuous.
Theorem3.5. For a functionf:(X, τ)→(Y , σ ), the following statements are equiv- alent:
(a) f:(X, τ)→(Y , σ )is slightly continuous;
(b) f:(X, τ)→(Y , σu)is clopen-continuous;
(c) f:(X, τ)→(Y , σu)is continuous;
(d) f:(X, τu)→(Y , σu)is continuous.
Proof. (a)⇒(b). Letx∈XandVbe an open set of(Y , σu)containingf (x). There exists a clopen setWof(Y , σ )such thatf (x)⊂W⊂V. Sincefis slightly continuous, there exists a clopen setU containingx such thatf (U )⊂W and hencef (U )⊂V.
This shows thatf (X, τ)→(Y , σu)is clopen-continuous.
(b)⇒(c). This is obvious.
(c)⇒(a). Letx∈X andV be a clopen set of(Y , σ ) containingf (x). ThenV is an open set of(Y , σu)and there existsU∈τcontainingxsuch thatf (U )⊂V. Therefore, f:(X, τ)→(Y , σ )is slightly continuous.
(b)⇒(d). Letx∈XandVany open set of(Y , σu)containingf (x). By (b) there exists a clopen subsetUof(X, τ)containingxsuch thatf (U )⊂V. SinceUis open in(X, τu), f (X, τu)→(Y , σu)is continuous.
(d)⇒(c). Sinceτu⊂τ, the proof is obvious.
Definition3.6. A function f:(X, τ)→(Y , σ ) is said to beβ-clopen-continuous (resp.,pre-clopen-continuous, semi-clopen-continuous) if for each pointxofXand each open setV containingf (x), there exists aβ-clopen (resp., pre-clopen, semi-regular) setUcontainingxsuch thatf (U )⊂V.
Theorem3.7. For a functionf:(X, τ)→(Y , σ ), the following statements are equiv- alent:
(a) f:(X, τ)→(Y , σ )is sl.β.c. (resp., slightly semi-continuous, faintly precontinuous);
(b) f (X, τ)→(Y , σu) is β-clopen continuous (resp., semi-clopen continuous, pre- clopen continuous);
(c) f (X, τ)→(Y , σu)isβ-continuous (resp., semi-continuous, precontinuous);
(d) f (X, τu)→(Y , σu)isβ-continuous (resp., semi-continuous, precontinuous).
Proof. The proof is similar to that ofTheorem 3.5and is thus omitted.
Corollary3.8(see Pal and Bhattacharyya [7]). A functionf :(X, τ)→(Y , σ ) is faintly precontinuous if and only iff−1(V )∈PO(X)for everyδ∗-open setV ofY.
4. Comparisons. In this section, we investigate the relationships between slightly β-continuous functions and other related functions. For this purpose, we will recall some definitions of functions.
Definition4.1. A functionf:X→Y is said to beweaklyβ-continuous[27] (resp., weakly semi-continuous[6],almost weakly continuous[16], orquasi precontinuous[25]) if for each pointx∈Xand each open setVcontainingf (x)there existsU∈SPO(X) (resp.,U∈SO(X),U∈PO(X)) containingxsuch thatf (U )⊂Cl(V ).
Definition4.2. A functionf:X→Y is said to becontra-β-continuous [13] (resp., contra-precontinuous) [13] iff−1(F )∈SPO(X)(resp.,f−1(F )∈PO(X)) for each closed setF ofY.
Definition4.3. A functionf:X→Y is said to beβ-quasi-irresolute[14] if for each pointx∈Xand eachV∈SO(Y )containingf (x)there existsU∈SPO(X, x)such that f (U )⊂Cl(V ).
A function is said to beβ-irresolute[18] if the preimages ofβ-open sets areβ-open.
It is obvious that a functionf :X→Y is β-irresolute if and only if for each point x∈X and eachV ∈SPO(Y , f (x)) there existsU∈SPO(X, x)such thatf (U )⊂V.
We give an interesting characterization of β-quasi-irresolute functions and make clear the fact thatβ-irresolute functions areβ-quasi-irresolute. A functionf:X→Y isβ-quasi-irresolute if and only if for each pointx∈Xand eachV∈SPO(Y , f (x)) there existsU∈SPO(X, x)such thatf (U )⊂Cl(V ). This follows from the fact that for eachβ-open setVofY, Cl(V )=Cl(Int(Cl(V )))and Cl(V )∈SO(Y ).
From the above definitions we obtain the following diagram:
β-quasi-irresolute
β-continuous weaklyβ-continuous weakly semi-continuous
contra-β-continuous slightlyβ-continuous slightly semi-continuous
contra-precontinuous faintly precontinuous slightly continuous
almost weakly continuous
Remark4.4. Slight semi-continuity and faint precontinuity are independent of each other as Examples4.5and4.6show.
Example 4.5. Let X = {a, b, c}, τ the indiscrete topology, and σ = {∅, X,{a}, {b, c}}. The identity functionf :(X, τ)→(X, σ )is precontinuous and faintly pre- continuous. But it is not slightly semi-continuous sincef−1({a})is not semi-open in (X, τ).
Example4.6. LetX= {a, b, c},τ= {∅,{a},{b},{a, b}, X}, andσ= {∅,{a},{b, c}, X}. Then the identity f :(X, τ)→(X, σ )is slightly semi-continuous by [23, Exam- ple 2.1] but not faintly precontinuous asf−1({a})is not preclosed in(X, τ).
Remark4.7. Contra-β-continuity andβ-continuity are independent of each other as Examples4.8and4.9show.
Example4.8. The identity function on the real line with the usual topology is con- tinuous and henceβ-continuous. But it is not contra-β-continuous since the preimage of any singleton is notβ-open.
Example4.9. LetX= {a, b}be the Sierpinski space by settingτ= {∅,{a}, X}and σ= {∅,{b}, X}. The identity functionf:(X, τ)→(Y , σ )is contra-continuous by [10, Example 2.5] and hence contra-β-continuous but notβ-continuous.
Definition4.10. A topological spaceXis said to be
(a) extremally disconnected (briefly E.D.) if the closure of each open set ofX is open inX,
(b) a PS-space[4] if every preopen set ofXis semi-open inX, (c) locally indiscrete[20] if every open set ofXis closed inX.
Theorem4.11. For a functionf:X→Y, the following properties hold:
(a)Iff is sl.β.c. andXis E.D., thenf is faintly precontinuous.
(b)Iffis sl.β.c. andXis a PS-space, thenfis slightly semi-continuous.
(c)Iffis sl.β.c. andXis an E.D. and PS-space, thenf is slightly continuous.
Proof. (a) Letx∈XandV∈CO(Y , f (x)). Now, putU=f−1(V ). SinceXis E.D., we haveU∈PO(X, x)by [4, Theorem 5.1] andf (U )⊂V. Therefore,f is faintly pre- continuous.
(b) SinceX is a PS-space, everyβ-open set of Xis semi-open by [4, Theorem 2.1]
and the result follows easily.
(c) LetV∈CO(Y ). Then by (a) and (b),f−1(V )is semi-regular and pre-clopen inX.
Sincef−1(V )is semi-closed and preopen, we have Int(Cl(f−1(V )))=f−1(V ). Since f−1(V ) is semi-open and preclosed, we have Cl(Int(f−1(V )))=f−1(V ). Therefore, f−1(V )∈CO(X)andf is slightly continuous.
Remark 4.12. We may define a function f :X →Y to be slightlyα-continuous iff−1(V )is α-open inX for every clopen setV of Y. However, it is known in [22, Lemma 3.1] that a subset isα-open if and only if it is semi-open and preopen. There- fore, by the proof forTheorem 4.11(c) eachα-open andα-closed set is clopen. Hence, slightα-continuity is equivalent to slight continuity.
Theorem4.13. For a functionf:(X, τ)→(Y , σ ), the following properties hold:
(a)Iff is sl.β.c. and(Y , σ )is E.D., thenfisβ-quasi-irresolute.
(b)Iffis sl.β.c. and(Y , σ )is ultra regular, thenf isβ-continuous.
(c)Iff is sl.β.c. and(X, τ)is a PS-space and(Y , σ ) is E.D., thenf is weakly semi- continuous.
(d)Iff is sl.β.c. and(Y , σ )is locally indiscrete, thenf is β-continuous and contra β-continuous.
Proof. (a) Let x ∈ X and V ∈ SO(Y ) containing f (x). Then we have Cl(V )= Cl(Int(V ))and hence Cl(V )is clopen in(Y , σ )since(Y , σ )is E.D. Sincef is sl.β.c., there existsU∈SPO(X, x)such thatf (U )⊂Cl(V ). Therefore,fisβ-quasi-irresolute.
(b) Since(Y , σ )is ultra regular,σu=σ and byTheorem 3.7the proof is obvious.
(c) Letx∈XandVany open set containingf (x). Then we have Cl(V )∈CO(Y )since (Y , σ )is E.D. Sincefis sl.β.c., there existsU∈SPO(X, x)such thatf (U )⊂Cl(V ). Since (X, τ)is a PS-space,U∈SO(X)by [4, Theorem 2.1], hencefis weakly semi-continuous.
(d) LetV be any open set of(Y , σ ). Since(Y , σ ) is locally indiscrete,V is clopen and hencef−1(V )isβ-open andβ-closed in(X, τ). Therefore,fisβ-continuous and contraβ-continuous.
Theorem4.14. For a functionf:X→Y, the following properties hold:
(a) Iffis sl.β.c.,Xis E.D. andYis locally indiscrete, thenfis contra-precontinuous.
(b) Iffis sl.β.c. andXandY are E.D., thenf is almost weakly continuous.
Proof. (a) LetF be any closed set ofY. ByTheorem 4.13(d),f is contra-β-contin- uous andf−1(F )∈SPO(X). SinceX is E.D.,f−1(F )∈PO(X)and hencef is contra- precontinuous.
(b) Letx∈XandVany open set containingf (x). Then we have Cl(V )∈CO(Y )since Y is E.D. Sincefis sl.β.c., there existsU∈SPO(X, x)such thatf (U )⊂Cl(V ). SinceX is E.D.,U∈PO(X), hencefis almost weakly continuous by [26, Theorem 3.1].
5. Properties. The composition of two slightlyβ-continuous functions need not be slightlyβ-continuous as shown by the following example due to Pal and Bhattacharyya [7].
Example5.1. LetX= {a, b, c},τ = {∅, X,{a}}, σ = {∅, X}, andθ= {∅, X,{a}, {b, c}}. Letf:(X, τ)→(X, σ )be the identity function andg:(X, σ )→(X, θ)a func- tion defined byg(a)=b,g(b)=c, andg(c)=a. Thenf andgare faintly precon- tinuous by [7, Example 4] and hence sl.β.c. However, the composition g◦f is not sl.β.c.
If f :X→Y is an open continuous function, thenf is β-irresolute and also the imagef (U )of eachβ-open set ofXisβ-open inY.
Theorem5.2. Letf:X→Y andg:Y→Zbe functions. Then (a) iffis sl.β.c. andgis slightly continuous, theng◦f is sl.β.c., (b) iffisβ-irresolute and g is sl.β.c., theng◦f is sl.β.c.,
(c) letf be an open continuous surjection. Then gis sl.β.c. if and only ifg◦f is sl.β.c.
Proof. (a) LetW ∈CO(Z). By the slight continuity of g, g−1(W )∈CO(Y ) and hencef−1(g−1(W ))=(g◦f )−1(W )∈SPO(X)sincef is sl.β.c. This shows thatg◦f is sl.β.c.
(b) Let W∈CO(Z). By the slightβ-continuity ofg,g−1(W )∈SPO(Y )and hence f−1(g−1(W ))=(g◦f )−1(W )∈SPO(X)sincef isβ-irresolute. This shows thatg◦f is sl.β.c.
(c) Letg be sl.β.c. Then, by (b)g◦f is sl.β.c. Conversely, letg◦f be sl.β.c. and W∈CO(Z). Then(g◦f )−1(W )∈SPO(X). Sincef is an open continuous surjection, f ((g◦f )−1(W ))=g−1(W )∈SPO(Y ). This shows thatgis sl.β.c.
Lemma 5.3(see Abd El-Monsef et al. [1]). LetX be a topological space and A, U subsets ofX. Then
(a) ifUisα-open inXandA∈SPO(X), thenA∩U∈SPO(U ), (b) ifA∈SPO(U )andU∈SPO(X), thenA∈SPO(X).
Theorem5.4. Let{Uγ:γ∈Γ}be anyα-open cover of a topological spaceX. A func- tionf:X→Yis sl.β.c. if and only if the restrictionf|Uγ:Uγ→Yis sl.β.c. for eachγ∈Γ.
Proof
Necessity. Let γ be an arbitrarily fixed index and Uγ an α-open set ofX. Let x∈UγandV∈CO(Y )containing(f|Uγ)(x)=f (x). Sincef is sl.β.c., there exists U∈SPO(X)containingxsuch thatf (U )⊂V. SinceUγisα-open inX, byLemma 5.3 x∈U∩Uγ∈SPO(Uγ)and(f|Uγ)(U∩Uγ)=f (U∩Uγ)⊂f (U )⊂V. This shows that f|Uγis sl.β.c.
Sufficiency. Letx∈XandV∈CO(Y )containingf (x). There exists aγ∈Γ such thatx∈Uγ. Sincef |Uγ :Uγ→Y is sl.β.c., there existsU∈SPO(Uγ)containing x such that(f|Uγ)(U )⊂V. By Lemma 5.3, U∈SPO(X)andf (U )⊂V. Therefore, f is sl.β.c.
Theorem5.5. A functionf :X→Y is sl.β.c. if the graph functiong:X→X×Y, defined byg(x)=(x, f (x))for eachx∈X, is sl.β.c.
Proof. Suppose thatgis sl.β.c. LetFbe a clopen set ofY. ThenX×Fis a clopen set ofX×Y. Sincegis sl.β.c.,g−1(X×F )=f−1(F )∈SPO(X). Therefore,f is sl.β.c.
Let{Xλ:λ∈Λ}and{Yλ:λ∈Λ}be two families of topological spaces with the same index set Λ. The product space of{Xλ:λ∈Λ}is denoted byΠ{Xλ:λ∈Λ} (or simplyΠXλ). Letfλ:Xλ→Yλbe a function for eachλ∈Λ. The product function f:ΠXλ→ΠYλis defined byf ({xλ})={fλ(xλ)}for each{xλ} ∈ΠXλ.
Theorem5.6. If a functionf:X→ΠYλis sl.β.c., thenPλ◦f:X→Yλis sl.β.c. for eachλ∈Λ, wherePλis the projection ofΠYλontoYλ.
Proof. LetVλbe any clopen set ofYλ. ThenPλ−1(Vλ)is clopen inΠYλand hence (Pλ◦f )−1(Vλ)=f−1(Pλ−1(Vλ))isβ-open inX. Therefore,Pλ◦fis sl.β.c.
Theorem5.7. If a functionf:ΠXλ→ΠYλ is sl.β.c., thenfλ:Xλ→Yλis sl.β.c. for eachλ∈Λ.
Proof. Let Vλ be any clopen set of Yλ. Then, Pλ−1(Vλ) is clopen in ΠYλ and f−1(Pλ−1(Vλ))=fλ−1(Vλ)×Π{Xα:α∈Λ− {λ}}. Sincef is sl.β.c.,f−1(Pλ−1(Vλ))is β- open inΠXλ. Since the projectionPλ ofΠXλontoXλis open continuous,fλ−1(Vλ)is β-open inXλand hencefλis sl.β.c.
Definition5.8. A topological spaceXis said to be
(a) β-Hausdorff [18] (resp.,ultra Hausdorff [30]) if every two distinct points ofX can be separated by disjointβ-open (resp., clopen) sets,
(b) β-regular[2] (resp.,ultra regular[12]) if each pair of a point and a closed set not containing the point can be separated by disjointβ-open (resp., clopen) sets, (c) β-normal [18] (resp.,ultra normal [30]) if every two disjoint closed sets ofX
can be separated byβ-open (resp., clopen) sets.
Theorem5.9. Letf:X→Y be a sl.β.c. injection. Then (a) ifY is ultra Hausdorff, thenXisβ-Hausdorff,
(b) ifY is ultra regular andf is open or closed, thenXisβ-regular, (c) ifY is ultra normal andf is closed, thenXisβ-normal.
Proof. (a) Letx1, x2 be two distinct points ofX. Then sincef is injective and Y is ultra Hausdorff, there existV1, V2∈CO(Y )such thatf (x1)∈V1, f (x2)∈V2, andV1∩V2= ∅. ByTheorem 3.2, xi∈f−1(Vi)∈SPO(X)fori=1,2 andf−1(V1)∩ f−1(V2)= ∅. ThusXisβ-Hausdorff.
(b) (i) Suppose thatf is open. Letx∈X andU be an open set containingx. Then f (U ) is an open set of Y containing f (x). SinceY is ultra regular, there exists a clopen setVsuch thatf (x)∈V⊂f (U ). Sincef is a sl.β.c. injection, by Theorem 3.1 x∈f−1(V )⊂Uandf−1(V )isβ-clopen inX. Therefore,Xisβ-regular. (ii) Suppose thatfis closed. Letx∈XandF be any closed set ofXnot containingx. Sincef is injective and closed,f (x)∉f (F )andf (F )is closed inY. By the ultra regularity ofY, there exists a clopen setVsuch thatf (x)∈V⊂Y−f (F ). Therefore,x∈f−1(V )and F⊂X−f−1(V ). ByTheorem 3.2,f−1(V )is aβ-clopen set inX. Thus,Xisβ-regular.
(c) LetF1,F2be disjoint closed subsets ofX. Sincefis closed and injective,f (F1) andf (F2)are disjoint closed subsets ofY. SinceYis ultra normal,f (F1)andf (F2)are separated by disjoint clopen setsV1andV2. Therefore, we obtain Fi⊂f−1(Vi)and f−1(Vi)∈SPO(X)fori=1,2 fromTheorem 3.2. Moreover,f−1(V1)∩f−1(V2)= ∅. ThusXisβ-normal.
A subset A of a topological space X is said to be semi pre β-closed if for each x∈X−Athere exists aβ-clopen setUcontainingxsuch thatU∩A= ∅.
Theorem5.10. Iff:X→Y is sl.β.c. andY is ultra Hausdorff, then (a) the graphG(f )offis semi preβ-closed in the product spaceX×Y,
(b) the set{(x1, x2):f (x1)=f (x2)}is semi preβ-closed in the product spaceX×X.
Proof. (a) Let(x, y)∈(X×Y )−G(f ). Theny≠f (x)and there exist clopen sets V andW such thaty∈V,f (x)∈W, andV∩W= ∅. Sincef is sl.β.c., there exists a β-clopen setUcontainingxsuch thatf (U )⊂W. Therefore, we obtainV∩f (U )= ∅ and hence(U×V )∩G(f )= ∅andU×V is aβ-clopen set ofX×Y. This shows that G(f )is semi preβ-closed inX×Y.
(b) SetA= {(x1, x2):f (x1)=f (x2)}. Let(x1, x2)∉A, thenf (x1)≠f (x2). SinceY is ultra Hausdorff, there existV1, V2∈CO(Y )containingf (x1),f (x2), respectively, such thatV1∩V2= ∅. Sincefis sl.β.c., there existβ-clopen setsU1,U2ofXsuch that xi∈Uiand f (Ui)⊂Vi fori=1,2.Thus,(x1, x2)∈U1×U2and(U1×U2)∩A= ∅.
Moreover,U1×U2isβ-clopen inX×XandAis semi preβ-closed inX×X.
A topological spaceXis said to beβ-connected[27] ifXcannot be expressed as the union of two disjoint nonemptyβ-open sets.
Theorem5.11. Iff:X→Y is a sl.β.c. surjection andXis β-connected, thenY is connected.
Proof. Assume thatY is not connected. Then there exist nonempty open setsV1
andV2such thatV1∩V2= ∅andV1∪V2=Y. Therefore,V1andV2are clopen sets ofY. Sincefis sl.β.c.,f−1(V1)andf−1(V2)areβ-open sets inX. Moreover, we have f−1(V1)∩f−1(V2)= ∅andf−1(V1)∪f−1(V2)=X. Sincef is surjective,f−1(V1)and f−1(V2)are nonempty. Therefore,Xis notβ-connected. This is a contradiction and henceY is connected.
Corollary5.12(see Popa and Noiri [27]). Iff:X→Y is a weaklyβ-continuous surjection andXisβ-connected, thenY is connected.
Corollary 5.13. If f : X →Y is a contra β-continuous surjection and X is β- connected, thenY is connected.
Acknowledgement. The author would like to thank the referee for his valuable suggestions which led to the improvement of this paper.
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Takashi Noiri: Department of Mathematics, Yatsushiro College of Technology, Yatsushiro, Kumamoto,866-8501, Japan
E-mail address:[email protected]
Mathematical Problems in Engineering
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