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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].
Xis denoted by CO(X, x)(resp., SPO(X, x), SPR(X, x)). The intersection of all semi-closed (resp., presemi-closed,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.
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
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},
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.
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
(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
(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(V
i)∈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.
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(V
1)∩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.
References
[1] M. E. Abd El-Monsef, S. N. El-Deeb, and R. A. Mahmoud,β-open sets andβ-continuous mapping, Bull. Fac. Sci. Assiut Univ. A 12 (1983), no. 1, 77–90.MR 87b:54002. Zbl 577.54008.
[2] M. E. Abd El-Monsef, A. N. Geaisa, and R. A. Mahmoud,β-regular spaces, Proc. Math. Phys. Soc. Egypt (1985), no. 60, 47–52.MR 89f:54042. Zbl 663.54015.
[3] M. E. Abd El-Monsef, R. A. Mahmoud, and E. R. Lashin,β-closure andβ-interior, J. Fac. Educ., Sec. A, Ain Shams Univ.10(1986), 235–245.Zbl 616.54001.
[4] T. Aho and T. Nieminen,Spaces in which preopen subsets are semiopen, Ricerche Mat.43 (1994), no. 1, 45–59.MR 96b:54058. Zbl 912.54030.
[5] D. Andrijevi´c,Semipreopen sets, Mat. Vesnik 38 (1986), no. 1, 24–32. MR 87j:54002. Zbl 604.54002.
[6] S. P. Arya and M. P. Bhamini,Some weaker forms of semicontinuous functions, Gan.ita33 (1982), no. 1-2, 124–134.CMP 910 778 001. Zbl 586.54017.
[7] M. Chandra Pal and P. Bhattacharyya,Faint precontinuous functions, Soochow J. Math.21 (1995), no. 3, 273–289.MR 96f:54016. Zbl 877.54011.
[8] S. G. Crossley and S. K. Hildbrand,Semi-closure, Texas J. Sci.22(1971), 99–112. [9] 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.
[10] J. Dontchev,Contra-continuous functions and stronglyS-closed spaces, Int. J. Math. Math. Sci.19(1996), no. 2, 303–310.MR 96m:54021. Zbl 840.54015.
[11] 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.
[12] R. L. Ellis,A non-Archimedean analogue of the Tietze-Urysohn extension theorem, Nederl. Akad. Wetensch. Proc. Ser. A70(1967), 332–333.MR 35#3636. Zbl 148.16402. [13] S. Jafari and T. Noiri,On contra precontinuous functions, in preparation.
[14] ,Onβ-quasi-irresolute functions, Mem. Fac. Sci. Kôchi Univ. Ser. A Math.21(2000), 53–62.CMP 1 744 539. Zbl 945.54013.
[15] R. C. Jain,The role of regularly open sets in general topology, Ph.D. thesis, Meerut Uni-versity, Meerut, 1980.
[17] N. Levine,Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly 70(1963), 36–41.MR 29#4025. Zbl 113.16304.
[18] R. A. Mahmoud and M. E. Abd El-Monsef,β-irresolute andβ-topological invariant, Proc. Pakistan Acad. Sci.27(1990), no. 3, 285–296.MR 92b:54001.
[19] A. S. Mashhour, M. E. Abd El-Monsef, and S. N. El-Deep,On precontinuous and weak precontinuous mappings, Proc. Math. Phys. Soc. Egypt (1982), no. 53, 47–53. MR 87c:54002. Zbl 571.54011.
[20] T. Nieminen,On ultrapseudocompact and related spaces, Ann. Acad. Sci. Fenn. Ser. A I Math.3(1977), no. 2, 185–205.MR 58#30964. Zbl 396.54009.
[21] O. Nj˙astad,On some classes of nearly open sets, Pacific J. Math.15 (1965), 961–970. MR 33#3245. Zbl 137.41903.
[22] T. Noiri, On α-continuous functions, ˇCasopis Pˇest. Mat. 109 (1984), no. 2, 118–126. MR 85e:54013. Zbl 544.54009.
[23] T. Noiri and G. I. Chae,A note on slightly semi-continuous functions, Bull. Calcutta Math. Soc.92(2000), no. 2, 87–92.CMP 1 822 377.
[24] T. M. Nour,Slightly semi-continuous functions, Bull. Calcutta Math. Soc.87(1995), no. 2, 187–190.CMP 1 352 299. Zbl 826.54008.
[25] R. Paul and P. Bhattacharyya,Quasi-pre-continuous functions, J. Indian Acad. Math.14 (1992), no. 2, 115–126.CMP 1 261 772. Zbl 871.54018.
[26] V. Popa and T. Noiri,Almost weakly continuous functions, Demonstratio Math.25(1992), no. 1-2, 241–251.MR 93f:54020. Zbl 789.54014.
[27] ,Weaklyβ-continuous functions, An. Univ. Timi¸soara Ser. Mat.-Inform.32(1994), no. 2, 83–92.MR 97g:54019. Zbl 864.54009.
[28] I. L. Reilly and M. K. Vamanamurthy,On super-continuous mappings, Indian J. Pure Appl. Math.14(1983), no. 6, 767–772.MR 85f:54024. Zbl 509.54007.
[29] A. R. Singal and D. S. Yadav,A generalization of semi-continuous mappings, J. Bihar Math. Soc.11(1988), 1–9.MR 90i:54031. Zbl 854.54019.
[30] R. Staum,The algebra of bounded continuous functions into a nonarchimedean field, Pa-cific J. Math.50(1974), 169–185.MR 49#5803. Zbl 296.46052.
Takashi Noiri: Department of Mathematics, Yatsushiro College of Technology, Yatsushiro, Kumamoto,866-8501, Japan
