Volume 2007, Article ID 75721,13pages doi:10.1155/2007/75721
Research Article
On b-I-Open Sets and b-I-Continuous Functions
Metin Akda˘gReceived 2 March 2007; Revised 23 May 2007; Accepted 30 August 2007 Recommended by Sehie Park
We studied some more properties ofb-I-open sets and obtained several characterizations ofb-I-continuous functions which are introduced by Caksu Guler and Aslim (2005). We also investigated their relationship with other types of functions.
Copyright © 2007 Metin Akda˘g. This is an open access article distributed under the Cre- ative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
One of the important and basic topics in the theory of classical point set topology and sev- eral branches of mathematics, which have been researched by many authors, is continuity of functions. This concept has been extended to the setting ofI-continuity of functions.
Jankovi´c and Hamlett [1,2] introduced the notion ofI-open sets in topological spaces.
Abd El-Monsef et al. [3] further investigatedI-open sets andI-continuous functions.
Dontchev [4] introduced the notion of pre-I-open sets and obtained a decomposition of I-continuity. The notion of semi-I-open sets to obtain decomposition of continuity was introduced by Hatir and Noiri [5,6]. In addition to this, Caksu Guler and Aslim [7] have introduced the notion ofb-I-sets andb-I-continuous functions. In the light of the above results, the purpose of this paper is to studyb-I-open sets andb-I-continuous functions and to obtain several characterizations and properties of these concepts.
2. Preliminaries
Throughout this paper, int(A) and Cl(A) denote the interior and closure ofA, respec- tively. An ideal is defined as a nonempty collection I of subsets of X satisfying the
following two conditions. (1) IfA∈I and B⊂A, thenB∈I. (2) If A∈I andB∈I, thenA∪B∈I. An ideal topological space is a topological space (X,τ) with an ideal I on X, and it is denoted by (X,τ,I). For a subset A⊂X, A∗(I)= {x∈X:U∩A /∈I for each neighborhood Uofx}is called the local function ofAwith respect to I and τ [8]. We simply writeA∗instead ofA∗(I) to be brief.X∗is often a proper subset of X. The hypothesisX=X∗[9] is equivalent to the hypothesisτ∩I=∅[10]. For every ideal topological space (X,τ,I), there exists a topologyτ∗(I), finer thanτ, generated by β(I,τ)= {U−I :U∈τand I∈I}, but in generalβ(I,τ) is not always a topology [1].
Additionally, Cl∗(A)=A∪(A)∗ defines a Kuratowski closure operator for τ∗(I). For a subsetA⊂X,Ais called∗-dense in itself [9] (resp.,τ∗-closed [1],∗-perfect [9]) if A⊂A∗(resp.,A∗⊂A,A=A∗). Given a space (X,τ,I) andA⊂X,Ais calledI-open if A⊂int(A∗) and a subsetKis calledI-closed if its complement isI-open [3].
3.b-I-open sets
First we will recall some definitions used in sequel.
Definition 3.1. A subsetSof a topological spaceXis said to be (a)α-open set [11] ifS⊂int(cl(int(S))),
(b) semiopen set [12] ifS⊂cl(int(S)), (c) preopen set [13] ifS⊂int(cl(S)), (d)β-open set [14] ifS⊂cl(int(cl(S))),
(e)b-open set (orγ-open [15]) [16] ifS⊂cl(int(S))∪int(cl(S)).
The class of all semiopen (preopen,α-open) sets in X will be denoted by SO(X,τ) (PO(X,τ),αO(X,τ)).
Definition 3.2. A subsetSof an ideal topological spaceXis said to be (a)α-I-open set [5] ifS⊂int(cl∗(int(S))),
(b) semi-I-open set [5] ifS⊂cl∗(int(S)), (c) pre-I-open set [3] ifS⊂int(cl∗(S)), (d)β-I-open set [5] ifS⊂cl∗(int(cl∗(S))), (e)I-open set [1] ifS⊂int(S∗),
(f)b-I-open set [7] ifS⊂cl∗(int(S))∪int(cl∗(S)).
The class of all semi-I-open (pre-I-open,α-open, andb-I-open) sets inXwill be de- noted by SIO(X,τ) (PIO(X,τ),αIO(X,τ), and BIO(X,τ)).
Proposition 3.3. For a subset of an ideal topological space, the following conditions hold:
(a) everyb-I-open set isb-open;
(b) every pre-I-open set isb-I-open [7];
(c) every semi-I-open set isb-I-open [7];
(d) SIO(X,τ)∪PIO(X,τ)⊂BIO(X,τ).
Proof. The proof is obvious.
Remark 3.4. For several sets defined above, we have the following implications:
α-open
open α-I-open semi-I-open semiopen
I-open pre-I-open b-I-open b-open
preopen
(3.1)
Example 3.5. Consider the setRof real numbers with the usual topology with idealI= {∅}and letS=[0, 1]∪((1, 2)∩Q), whereQ stands for the set of rational numbers.
ThenSisb-I-open set but neither semi-I-open nor pre-I-open. On the other hand, let T=[0, 1)∩Q. ThenTis notb-I-open.
Example 3.6. Let (R,τ) be the real numbers with the usual topology andI the ideal of all finite sets ofR. LetQbe the set of all rationals. SinceQ∗(I)=R, thenQisb-I-open.
Since cl∗(intQ)=∅,Qis not semi-I-open.
Example 3.7. LetX= {a,b,c,d}be the topological space by setting
τ=
X,∅,{b},{c,d},{b,c,d}
, I=
{c},{d},{c,d},∅. (3.2)
ThenA= {a,b}is not pre-I-open but it isb-I-open.
Proposition 3.8. LetSbe ab-I-open set such that intS=∅. ThenSis pre-I-open set.
Proof. Since S⊂cl∗(intS)∪int(cl∗(S))=cl∗(∅)∪int(cl∗(S))=int(cl∗(S)), then S is
pre-I-open.
Lemma 3.9. LetAandBbe subsets of a space (X,τ,I) [1]. Then (1) ifA⊂B, thenA∗⊂B∗;
(2) ifU∈τ, thenU∩A∗⊂(U∩A)∗.
Proposition 3.10. Let (X,τ,I) be an ideal topological space andA,Bsubsets ofX.
(a) IfUα∈BIO(X,τ) for eachα∈Δ, then∪{Uα:α∈Δ} ∈BIO(X,τ).
(b) IfA∈BIO(X,τ) andB∈τ, thenA∩B∈BIO(X,τ) [7].
Proof. (a) SinceUα∈BOI(X,τ), we haveUα⊂cl∗(int(Uα))∪int(cl∗(Uα)) for eachα∈ Δ. Then by usingLemma 3.9, we have
α∈Δ
Uα⊂
α∈Δ
cl∗intUα
∪intcl∗Uα
⊂
α∈Δ
intUα∪
intUα∗∪
intUα∪
Uα∗
⊂ int
α∈Δ
Uα
∪
α∈Δ
intUα∗
∪ int
α∈Δ
Uα
∪
α∈Δ
Uα∗
⊂ int
α∈Δ
Uα
∪
int
α∈Δ
Uα
∗
∪ int
α∈Δ
Uα
∪
α∈Δ
Uα
∗
⊂cl∗ int
α∈Δ
Uα
∪ int cl∗
α∈Δ
Uα
.
(3.3)
Henceα∈ΔUαisb-I-open.
(b) LetA∈BIO(X,τ) andB∈τ. ThenA⊂cl∗(intA)∪int(cl∗(A)) and A∩B⊂
cl∗(intA)∪intcl∗(A)∩B
=
cl∗(intA)∩B∪
intcl∗(A)∩B
=
(intA)∪(intA)∗∩B∪
intA∪(A)∗∩B
⊂
(intA)∩B∪
(intA)∩B∗∪
int(A∩B)∪(A∗∩B)
⊂
int(A∩B)∗]∪
int(A∩B)∪
int(A∩B)∗∪(A∩B)
=cl∗int(A∩B)∪intcl∗(A∩B).
(3.4)
This shows thatA∩B∈BIO(X,τ).
Definition 3.11. A subsetAof a space (X,τ,I) is said to beb-I-closed if its complement is b-I-open.
Theorem 3.12. If a subsetAof a space (X,τ,I) isb-I-closed, then int(cl∗(A))∩cl∗(intA)
⊂A.
Proof. SinceAisb-I-closed,X−A∈BIO(X,τ) and sinceτ∗(I) is finer thanτ, we have X−A⊂cl∗int(X−A)∪intcl∗(X−A)⊂clint(X−A)∪intcl(X−A)
= X−
intcl(A)∪ X−
cl(intA)
⊂ X−
intcl∗(A)∪ X−
cl∗(intA)
=X−
intcl∗(A)∩
cl∗(intA).
(3.5)
Therefore, we obtain
intcl∗(A)∩
cl∗(intA)⊂A. (3.6)
Corollary 3.13. Let A be a subset of (X,τ,I) such that X −[int(cl∗(A))] = cl∗(int(X−A)) andX−[cl∗(intA)]=int(cl∗(X−A)). ThenAisb-I-closed if and only if int(cl∗(A))∩cl∗(intA)⊂A.
Proof. Necessity. This is an immediate consequence ofTheorem 3.12.
Sufficiency. Let int(cl∗(A))∩cl∗(intA)⊂A. Then X−A⊂X−
intcl∗(A)∩cl∗(intA)
⊂ X−
intcl∗(A)∪ X−
cl∗(intA)
=cl∗int(X−A)∪intcl∗(X−A).
(3.7)
ThusX−Aisb-I-open and soAisb-I-closed.
If (X,τ,I) is an ideal topological space andAis a subset ofX, we denote byτ|A. the relative topology onAandI|A= {A∩I:I∈I}is obviously an ideal onA.
Lemma 3.14 (see [1]). Let (X,τ,I) be an ideal topological space andA,Bsubsets ofXsuch thatB⊂A. ThenB∗(τ|A,I|A)=B∗(τ,I)∩A.
Theorem 3.15. Let (X,τ,I) be an ideal topological space. IfU∈τandW∈BIO(X,τ), thenU∩W∈BIO(U,τ|U,I|U).
Proof. SinceU is open, we have intUA=intAfor any subsetAofU. By using this fact andLemma 3.14, we have
U∩W⊂U∩
cl∗int(W)∪intcl∗(W)
= U∩
(intW)∪(intW)∗∪ U∩
int(W∪W∗)
⊂ U∩
U∩(intW)∪U∩(intW)∗∪ U∩
U∩
int(W∪W∗)
⊂ U∩
U∩(intW)∪(U∩intW)∗
∪ U∩
U∩
int(W∪W∗)
⊂ U∩
intU(U∩W)∪
U∩intU(U∩W)∗
∪ U∩
int(U∩W)∪(U∩W)∗
=
intU(U∩W)∪
intU(U∩W)∗(τ|U.,I|U.)
∪ U∩
int(U∩W)∪(U∩W)∗
=cl∗UintU(U∩W)∪ intU
cl∗U(U∩W).
(3.8)
This shows thatU∩W∈BIO(U,τ|U·,I|U·).
Proposition 3.16 (see [7]). For an ideal topological space (X,τ,I) andA⊂X, we have the following.
(1) IfI=∅, thenAisb-I-open if and only ifAisb-open.
(2) IfI=P(x), thenAisb-I-open if and only ifA∈τ.
(3) IfI=N, thenAisb-I-open if and only ifAisb-open, whereN is the ideal of all nowhere dense sets.
Lemma 3.17 (see [1]). Let (X,τ,I) be an ideal topological space and letA⊂X. Then if U∈τ,U∩A∗=U∩(U∩A)∗⊂(U∩A)∗.
Proposition 3.18. Let (X,τ,I) be an ideal topological space withΔbeing an arbitrary index set. Then
(1) ifA∈BIO(X,τ) andB∈τα, thenA∩B∈BO(X,τ);
(2) ifA∈PIO(X,τ) andB∈SIO(X,τ), thenA∩B∈SO(A);
(3) ifA∈PIO(X,τ) andB∈SIO(X,τ), thenA∩B∈PO(B).
Proof. (1) Since intersection ofb-open andα-set is always ab-open set [16, Proposition 2.4], then the claim is clear due toProposition 3.3.
(2)-(3) It was proved in [17] that the intersection of a preopen and a semiopen set is a preopen subset of the semiopen set and a semiopen subset of the preopen set. Thus the
claim follows from [5, Proposition 2.5] and [6].
Proposition 3.19. Eachb-I-open subset which isτ∗-closed is semi-I-closed.
Proof. LetAbeb-I-open andτ∗-closed set. Then A⊂intcl∗(A)∪cl∗(intA)=intA∪
intA∪(intA)∗
=intA∪(intA)∗=cl∗(intA). (3.9)
Definition 3.20. IfSis a subset of a space (X,τ,I), then
(a) theb-I-closure ofS, denoted by cl∗b(S), is the smallestb-I-closed set containing S;
(b) theb-I-interior ofS, denoted by intbI(S), is the largestb-I-open set contained in S.
Lemma 3.21. (1) LetAbe a subset of a space (X,τ,I). ThenAisb-I-closed if and only if cl∗b(A)=A.
(2) LetBbe a subset of a space (X,τ,I). ThenAisb-I-open if and only if intbI(B)=B.
Proposition 3.22. LetA,Bbe subsets of a space (X,τ,I) such thatAisb-I-open andBis b-I-closed inX. Then there exist ab-I-open setHand ab-I-closed setKsuch thatA∩B⊂K andH⊂A∪B.
Proof. LetK=cl∗b(A)∩B andH=A∪intbI(B). Then,K isb-I-closed andH isb-I- open. A⊂cl∗b(A) implies A∩B⊂cl∗b(A)∩B=K and intbI(B)⊂B implies A∪
intbI(B)=H⊂A∪B.
Definition 3.23. (1) A subsetSof a space (X,τ,I) is calledb-dense if clb(S)=X, where clb(S) is the smallestb-closed set containingS[15].
(2) A subsetSof a space (X,τ,I) is calledb-I-dense if cl∗b(S)=X.
Remark 3.24. Everyb-I-dense subset in a space (X,τ,I) isb-dense.
4.b-I-continuous mappings
Definition 4.1. (a) A functionf : (X,τ)→(Y,σ) is calledb-continuous (orγ-continuous) if the inverse image of each open set inYisb-open set inX[15].
(b) A function f : (X,τ)→(Y,σ) is called precontinuous if the inverse image of each open set inY is preopen set inX[13].
(c) A function f : (X,τ,I)→(Y,σ) is called pre-I-continuous if the inverse image of each open set inYis pre-I-open set inX[18].
(d) A functionf : (X,τ)→(Y,σ) is called semicontinuous if the inverse image of each open set inY is semiopen set inX[12].
(e) A function f : (X,τ,I)→(Y,σ) is called semi-I-continuous if the inverse image of each open set inYis semi-I-open set inX[5].
(f) A function f : (X,τ)→(Y,σ) is calledα-continuous (orγ-continuous) if the in- verse image of each open set inYisα-open set inX[14].
(g) A function f : (X,τ,I)→(Y,σ) is calledα-I-continuous if the inverse image of each open set inYisα-I-open set inX[5].
(h) A function f : (X,τ,I)→(Y,σ) is calledb-I-continuous if the inverse image of each open set inYisb-I-open set inX[7].
Remark 4.2 (see [7, Propositions 6 and 7]). (1)b-I-continuity impliesb-continuity.
(2) semi-I-continuity impliesb-I-continuity.
(3) pre-I-continuity impliesb-I-continuity.
Definition 4.3 (see [19]). LetAbe a subset of a space (X,τ,I).
Then the set∩{U∈τ:A⊂U}is called the kernel ofAand denoted by Ker(A).
Lemma 4.4 (see [20]). LetAbe a subset of a space (X,τ), then
(a)x∈Ker(A) if and only ifA∩F=∅for any closed subsetFofXwithx∈F;
(b)A⊂Ker(A) andA=Ker(A) ifAis open inX;
(c) ifA⊂B, then Ker(A)⊂Ker(B).
Definition 4.5. LetNbe a subset of a space (X,τ,I) and letx∈X. ThenNis calledb-I- neighborhood ofx, if there exists ab-I-open setUcontainingxsuch thatU⊂N.
Theorem 4.6. The following statements are equivalent for a function f : (X,τ,I)→(Y,σ):
(a) f isb-I-continuous;
(b) for eachx∈Xand each open setVinYwith f(x)∈V, there exists ab-I-open set Ucontainingxsuch thatf(U)⊂V;
(c) for each x∈X and each open set V in Y with f(x)∈V, f−1(V) is a b-I- neighborhood ofx;
(d) the inverse image of each closed set in (Y,σ) isb-I-closed;
(e) for every subsetAofX,f(intbI(A))⊂Ker(f(A));
(f) for every subsetBofY, intbI(f−1(B))⊂ f−1(Ker(B)).
Proof. (a)⇒(b). Letx∈X and letV be an open set inY such that f(x)∈V. Since f is b-I-continuous, f−1(V) isb-I-open. By puttingU= f−1(V) which is containingx, we have f(U)⊂V.
(b)⇒(c). LetV be an open set inY and let f(x)∈V. Then by (b), there exists ab- I-open setUcontainingxsuch that f(U)⊂V. Sox∈U⊂ f−1(V). Hence f−1(V) is a b-I-neighborhood ofx.
(c)⇒(a). LetV be an open set inY and let f(x)∈V. Then by (c), f−1(V) is ab-I- neighborhood ofx. Thus for eachx∈ f−1(V), there exists ab-I-open setUxcontaining xsuch thatx∈Ux⊂f−1(V). Hencef−1(V)⊂
x∈f−1(V)Uxand sof−1(V)∈BOI(X,τ).
(a)⇔(d). It is obvious.
(a)⇒(e). LetAbe any subset ofX. Suppose thaty /∈Ker(f(A)). Then, byLemma 4.4, there exists a closed subset F of Y such that y∈F and f(A)∩F=∅. Thus we have A∩f−1(F)=∅and (intbI(A))∩f−1(F)=∅. Therefore, we obtain f(intbI(A))∩F=∅ andy /∈ f(intbI(A)). This implies that f(intbI(A))⊂Ker(A).
(e)⇒(f). LetBbe any subset ofY. By (e) andLemma 4.4, we havef(intbI(f−1(B)))⊂ Ker(f(f−1(B)))⊂Ker(B) and intbI(f−1(B))⊂ f−1(Ker(B)).
(f)⇒(a). Let V be an open set of Y. Then by Lemma 4.4 and (f), we have intbI(f−1(V))⊂ f−1(Ker(V))= f−1(V) and intbI(f−1(V))= f−1(V). This shows that
f−1(V) isb-I-open.
The following examples show thatb-I-continuous functions do not need to be semi- I-continuous and pre-I-continuous, andb-continuous function does not need to beb-I- continuous.
Example 4.7. LetX=Y= {a,b,c,d}be the topological space by settingτ=σ= {{a}, {d},{a,d},X,∅}, andI= {∅,{c}}onX.
Define a function f : (X,τ,I)→(Y,σ) as follows: f(a)= f(c)=dand f(b)=f(d)= b. Then f isb-I-continuous but it is not pre-I-continuous.
Example 4.8. Let (X,τ) be the real line with the indiscrete topology and (Y,σ) the real line with the usual topology. Then the identity function f : (X,τ,P(X))→(Y,σ) is b- continuous but notb-I-continuous.
Example 4.9. LetX=Y= {a,b,c}be the topological space by settingτ=σ= {X,∅,{a, b}}andI= {{c},∅}. Define a functionf : (X,τ,I)→(Y,σ) as follows:f(a)=a,f(b)=c, and f(c)=b. Then f isb-I-continuous but not semi-I-continuous.
Proposition 4.10. Let f : (X,τ,I)→(Y,σ,J) andg: (Y,σ,J)→(Z,ν) be two functions, whereIandJ are ideals onXandY, respectively. Theng◦f isb-I-continuous if f isb-I- continuous andgis continuous.
Proof. The proof is clear.
Theorem 4.11. Let f : (X,τ,I)→(Y,σ) beb-I-continuous andU∈τ. Then the restriction f|U: (U,τ|U,I|U)→(Y,σ) isb-I-continuous.
Proof. LetV be any open set of (Y,σ). Since f isb-I-continuous, f−1(V)∈BIO(X,τ) and byTheorem 3.15, (f|U)−1(V)= f−1(V)∩U∈BIO(U,I|U). This shows that f|U: (U,τ|U,I|U)→(Y,σ) isb-I-continuous.
Theorem 4.12. Let f : (X,τ,I)→(Y,σ,J) be a function and let{Uα:α∈Δ}be an open cover ofX. If the restriction function f|Uαisb-I-continuous for eachα∈Δ, then f isb-I- continuous.
Proof. The proof is similar to that ofTheorem 4.11.
Theorem 4.13. A function f : (X,τ,I)→(Y,σ) isb-I-continuous if and only if the graph functiong:X→X×Ydefined byg(x)=(x,f(x)) for eachx∈Xisb-I-continuous.
Proof. Necessity. Let f beb-I-continuous. Now letx∈X and letW be any open set in X×Y containingg(x)=(x,f(x)). Then there exists a basic open set U×V such that g(x)⊂U×V ⊂W. Since f isb-I-continuous, there exists ab-I-open setU1 inXsuch thatx∈U1⊂Xand f(U1)⊂V. ByProposition 3.10,U1∩U∈BOI(X,τ) andU1∩U⊂ U, theng(U1∩U)⊂U×V⊂W. This shows thatgisb-I-continuous.
Sufficiency. Suppose thatgisb-I-continuous and letVbe open set inYcontainingf(x).
ThenX×V is open set inX×Y and by theb-I-continuity ofg, there exists ab-I-open setUcontainingxsuch thatg(U)⊂X×V. Therefore, we obtain f(U)⊂V. This shows
that f isb-I-continuous.
Theorem 4.14. Let{Xα:α∈Δ}be any family of ideal topological spaces. If f : (X,τ,I)→ (α∈ΔXα,σ) is ab-I-continuous function, thenPα◦f :X→Xαisb-I-continuous for each α∈Δ, wherePαis the projection ofXαontoXα.
Proof. We will consider a fixedα◦∈Δ. LetGα0be an open set ofXα0. Then (Pα0)−1(Gα0) is open set inα=α0Xα. Since f isb-I-continuous, f−1((Pα0)−1(Gα0))=(Pα0◦f)−1(Gα0)
isb-I-open inX. ThusPα0◦f isb-I-continuous.
Lemma 4.15 (see [21]). For any function f : (X,τ,I)→(Y,σ), f(I) is an ideal onY. Definition 4.16 (see [21]). An ideal topological space (X,τ,I) is said to beI-compact if for everyI-open cover{Wα:α∈Δ}ofX, there exists a finite subsetΔ◦ ofΔsuch that (X− ∪{Wα:α∈Δ◦})∈I.
Definition 4.17. An ideal topological space (X,τ,I) is said to beb-I-compact if for everyb- I-open cover{Wα:α∈Δ}ofX, there exists a finite subsetΔ◦ofΔsuch that (X− ∪{Wα: α∈Δ◦})∈I.
Theorem 4.18. The image of ab-I-compact space under ab-I-continuous surjective func- tion is f(I)-compact.
Proof. Let f : (X,τ,I)→(Y,σ) be ab-I-continuous surjection and{Vα:α∈Δ}be an open cover ofY. Then{f−1(Vα) :α∈Δ}is ab-I-open cover ofXdue to our assump- tion on f. Since X isb-I-compact, then there exists a finite subset Δ◦ of Δsuch that (X− ∪{f−1(Vα) :α∈Δ◦})∈I. Therefore (Y− ∪{Vα:α∈Δ◦})∈f(I) which shows that
(Y,σ,f(I)) is f(I)-compact.
Definition 4.19. An ideal topological space (X,τ,I) is said to beb-I-normal if for each pair of nonempty disjoint closed sets ofX, it can be separated by disjointb-I-open sets.
Definition 4.20. An ideal topological space (X,τ,I) is said to beb-I-connected ifXis not the union of two disjointb-I-open subsets ofX.
Definition 4.21 (see [22]). A topological space (X,τ) is said to be ultra normal if for each pair of nonempty disjoint closed sets ofX, it can be separated by disjoint clopen sets.
Theorem 4.22. If f : (X,τ,I)→(Y,σ) is ab-I-continuous, closed injection andY is nor- mal, thenXisb-I-normal.
Proof. LetF1andF2be disjoint closed subsets ofX. Since f is closed and injective, f(F1) and f(F2) are disjoint closed subsets ofY. SinceY is normal, f(F1) and f(F2) are sep- arated by disjoint open setsV1andV2, respectively. HenceF1⊂f−1(V1),F2⊂ f−1(V2), f−1(V1)∈BIO(X,τ), f−1(V2)∈BIO(X,τ), and f−1(V1)∩f−1(V2)=∅. ThusXisb-I-
normal.
Corollary 4.23. If f : (X,τ,I)→(Y,σ) is ab-I-continuous, closed injection andYis ultra normal, thenXisb-I-normal.
Theorem 4.24. Ab-I-continuous image of ab-I-connected space is connected.
Proof. Let f : (X,τ,I)→(Y,σ) be ab-I-continuous function of ab-I-connected space X onto a topological space Y. If possible, letY be disconnected. LetAand Bform a disconnected set ofY. ThenAandBare clopen andY=A∪B, whereA∩B=∅. Since f isb-I-continuous,X=f−1(Y)= f−1(A∪B), where f−1(A) andf−1(B) are nonempty b-I-open sets inX. Also f−1(V1)∩f−1(V2)=∅. HenceXis non-b-I-connected, which
is a contradiction. Therefore,Yis connected.
Definition 4.25. A function f : (X,τ,I)→(Y,σ,J) is calledb-I-open (resp.,b-I-closed) if for eachU∈τ(resp., closed setF), f(U) (resp., f(F)) isb-J-open (resp.,b-J-closed).
Remark 4.26. Everyb-I-open (resp.,b-I-closed) function isb-open (resp.,b-closed) and the converses are false in general.
Example 4.27. LetX= {a,b,c},τ1= {X,∅,{b,c}},τ2= {X,∅,{a,b},{b},{a}}, andI= {{a},∅}. Then the identity functionf : (X,τ1)→(X,τ2,I) isb-open but notb-I-open.
Example 4.28. Let X= {a,b,c},τ1= {X,∅,{a}}, τ2= {X,∅,{b,c},{b},{c}}, and I= {{c},∅}. Define a function f : (X,τ1)→(X,τ2,I) as follows: f(a)=a,f(b)= f(c)=b.
Then, f isb-closed but notb-I-closed.
Definition 4.29. (a) A function f : (X,τ,I)→(Y,σ,J) is called semi-I-open (resp., semi- I-closed) if for eachU∈τ(resp., closed setF), f(U) (resp., f(F)) is semi-J-open (resp., semi-J-closed) [6] .
(b) A function f : (X,τ,I)→(Y,σ,J) is called pre-I-open (resp., pre-I-closed) if for eachU∈τ(resp., closed setF), f(U) (resp., f(F)) is pre-J-open (resp., pre-J-closed).
(c) A function f : (X,τ,I)→(Y,σ,J) is calledα-I-open (resp.,α-I-closed) if for each U∈τ(resp.,Uis closed), f(U) isα-I-open (resp.,α-I-closed).
Remark 4.30. (1) Every semi-I-open (resp., semi-I-closed) function isb-I-open (resp., b-I-closed); (2) every pre-I-open (resp., pre-I-closed) function isb-I-open (resp.,b-I- closed).
Theorem 4.31. A function f : (X,τ,I)→(Y,σ,J) isb-I-open if and only if for eachx∈X and each neighborhoodU ofx, there existsV∈BJO(Y,σ) containing f(x) such thatV⊂
f(U).
Proof. Suppose that f is ab-I-open function. For each x∈Xand each neighborhood U of x, there exists U◦∈τ such thatx∈U◦⊂U. Since f isb-I-open, V = f(U◦)∈ BJO(Y,σ) and f(x)∈V ⊂ f(U). Conversely, letU be an open set of (X,τ). For each x∈U, there exists Vx∈BIO(X,τ) such that f(x)∈Vx⊂ f(U). Therefore we obtain f(U)= ∪{Vx:x∈U} and hence by Proposition 3.10, f(U)∈BJO(Y,σ). This shows
that f isb-I-open.
Theorem 4.32. Letf : (X,τ,I)→(Y,σ,J) beb-I-open (resp.,b-I-closed). IfWis any subset ofYandFis a closed (resp., open) set ofXcontainingf−1(W), then there exists ab-I-closed (resp.,b-I-open) subsetHofY containingWsuch that f−1(H)⊂F.
Proof. Suppose that f is ab-I-open function. LetW be any subset ofY andF a closed subset of X containing f−1(W). ThenX−F is open and since f isb-I-open, f(X− F) is b-I-open. Hence H=Y− f(X−F) is b-I-closed. It follows from f−1(W)⊂F thatW⊂H. Moreover, we obtain f−1(H)⊂F. For ab-I-closed function, we can prove
Theorem 4.32similarly.
Theorem 4.33. For any bijective functionf : (X,τ)→(Y,σ,J), the following are equivalent:
(i) f−1: (Y,σ,J)→(X,τ) isb-I-continuous;
(ii) f isb-I-open;
(iii) f isb-I-closed.
Proof. It is straightforward.
Definition 4.34 ([4]). A function f : (X,τ,I)→(Y,σ,J) is called∗-I-continuous if the preimage of every open set in (Y,σ) is∗-dense in itself.
Proposition 4.35. For a subsetA⊂(X,τ,I), if the condition (int(A∗))∗⊂int(A∗) holds, then the following are equivalent:
(1)AisI-open;
(2)Aisb-I-open and∗-dense in itself.
Proof. (1)⇒(2) Let Abe an I-open subset of (X,τ,I). ThenA⊂int(A∗)⊂A∗, which shows thatAis∗-dense in itself. SinceAisI-open, thenA is pre-I-open and soA⊂ int(cl∗(A))⊂int(cl∗(A))∪cl∗(intA). ThusAisb-I-open.
(2)⇒(1) LetAbe ab-I-open and∗-dense in itself.
Then since (int(A∗))∗⊂int(A∗),A⊂int(cl∗(A))∪cl∗(intA)=int(A∪A∗)∪(intA∪ (intA)∗)⊂int(A∗)∪int(A)∪(int(A))∗=int(A∗)∪(int(A))∗=int(A∗).
Proposition 4.36. If a function f : (X,τ,I)→(Y,σ) isI-continuous and if (int(A∗))∗⊂ int(A∗) for each subsetAofX, thenf isb-I-continuous and∗-I-continuous.
Proof. FromProposition 4.35, the proof is clear.
Definition 4.37. A space (X,τ) is called
(1)b-space if everyb-open set ofXis open inX[23],
(2) submaximal if every dense set ofXis open inX, and equivalently, if every pre- open set is open,
(3) extremally disconnected if the closure of every open set ofXis open inX.
Corollary 4.38. If a functionf : (X,τ,I)→(Y,σ) is continuous, thenf isb-I-continuous.
Corollary 4.39. If (X,τ) isb-space, then for any idealIonX, BIO(X,τ)=BO(X,τ)=τ.
Corollary 4.40 ([18]). If (X,τ) is submaximal, then for any idealI onX, PIO(X,τ)= PO(X,τ)=τ.
Corollary 4.41. If (X,τ) isb-space, then for any idealIonX, BIO(X,τ)=BO(X,τ)= PIO(X,τ)=PO(X,τ)=τ.
Corollary 4.42. If (X,τ) is extremally disconnected and submaximal, then for any idealI onX, PIO(X,τ)=SIO(X,τ)=SO(X,τ)=PO(X,τ)=αO(X,τ)=αIO(X,τ)=τ.
Corollary 4.43. If (X,τ) isb-space, then for any idealIonX, BIO(X,τ)=BO(X,τ)= SIO(X,τ)=SO(X,τ)=τ.
Corollary 4.44. If (X,τ) isb-space, then for any idealIonX, BIO(X,τ)=BO(X,τ)= PIO(X,τ)=SIO(X,τ)=SO(X,τ)=PO(X,τ)=αO(X,τ)=αIO(X,τ)=τ.
Corollary 4.45. Let f : (X,τ,I)→(Y,σ) be a function and let (X,τ) beb-space, then the following are equivalent:
(1) f isb-I-continuous, (2) f isb-continuous, (3) f is pre-I-continuous, (4) f is precontinuous, (5) f is semi-I-continuous, (6) f is semicontinuous, (7) f isα-I-continuous, (8) f isα-continuous, (9) f is continuous.
Remark 4.46. Forb-I-open,b-open, semi-I-open, semiopen, pre-I-open, pre-open,α-I- open, and open functions, we have similar corollary if (X,τ) isb-space.
References
[1] D. Jankovi´c and T. R. Hamlett, “New topologies from old via ideals,” The American Mathematical Monthly, vol. 97, no. 4, pp. 295–310, 1990.
[2] D. Jankovi´c and T. R. Hamlett, “Compatible extensions of ideals,” Unione Matematica Italiana.
Bollettino. B. Serie VII, vol. 6, no. 3, pp. 453–465, 1992.
[3] M. E. Abd El-Monsef, E. F. Lashien, and A. A. Nasef, “OnI-open sets andI-continuous func- tions,” Kyungpook Mathematical Journal, vol. 32, no. 1, pp. 21–30, 1992.
[4] J. Dontchev, “On pre-Iopen sets and a decomposition ofIcontinuity,” Banyan Mathematical Journal, vol. 2, 1996.
[5] E. Hatir and T. Noiri, “On decompositions of continuity via idealization,” Acta Mathematica Hungarica, vol. 96, no. 4, pp. 341–349, 2002.
[6] E. Hatir and T. Noiri, “On semi-I-open sets and semi-I-continuous functions,” Acta Mathemat- ica Hungarica, vol. 107, no. 4, pp. 345–353, 2005.
[7] A. Caksu Guler and G. Aslim, “b-I-open sets and decomposition of continuity via idealization,”
Proceedings of Institute of Mathematics and Mechanics. National Academy of Sciences of Azerbai- jan, vol. 22, pp. 27–32, 2005.
[8] R. Vaidyanathaswamy, “The localisation theory in set-topology,” Proceedings of the Indian Acad- emy of Sciences, vol. 20, pp. 51–61, 1944.
[9] E. Hayashi, “Topologies defined by local properties,” Mathematische Annalen, vol. 156, pp. 205–
215, 1964.
[10] P. Samuels, “A topology formed from a given topology and ideal,” Journal of the London Mathe- matical Society, vol. 10, no. 4, pp. 409–416, 1975.
[11] O. Nj˙astad, “On some classes of nearly open sets,” Pacific Journal of Mathematics, vol. 15, pp.
961–970, 1965.
[12] N. Levine, “Semi-open sets and semi-continuity in topological spaces,” The American Mathe- matical Monthly, vol. 70, no. 1, pp. 36–41, 1963.
[13] A. S. Mashhour, M. E. Abd El-Monsef, and S. N. El-Deep, “On precontinuous and weak pre- continuous mappings,” Proceedings of the Mathematical and Physical Society of Egypt, no. 53, pp.
47–53, 1982.
[14] A. S. Mashhour, I. A. Hasanein, and S. N. El-Deeb, “α-continuous andα-open mappings,” Acta Mathematica Hungarica, vol. 41, no. 3-4, pp. 213–218, 1983.
[15] A. A. El-Atik, “A study of some types of mappings on topological spaces,” M. Sci. thesis, Tanta University, Tanta, Egypt, 1997.
[16] D. Andrijevi´c, “Onb-open sets,” Matematichki Vesnik, vol. 48, no. 1-2, pp. 59–64, 1996.
[17] T. Noiri, “Hyperconnectedness and pre-open sets,” Revue Roumaine de Math´ematiques Pures et Appliqu´ees, vol. 29, no. 4, pp. 329–334, 1984.
[18] J. Dontchev, “Idealization of Ganster-Reilly decomposition theorems,” 1999, http://arxiv.org/abs/Math.GN/9901017.
[19] M. Mrˇsevi´c, “On pairwiseR0and pairwiseR1bitopological spaces,” Bulletin Math´ematique de la Soci´et´e des Sciences Math´ematiques de la R´epublique Socialiste de Roumanie. Nouvelle S´erie, vol. 30(78), no. 2, pp. 141–148, 1986.
[20] S. Jafari and T. Noiri, “Contra-super-continuous functions,” Annales Universitatis Scientiarum Budapestinensis, vol. 42, pp. 27–34, 1999.
[21] R. L. Newcomb, “Topologies which are compact modulo an ideal,” Ph.D. dissertation, University of California, Santa Barbara, Calif, USA, 1967.
[22] R. Staum, “The algebra of bounded continuous functions into a nonarchimedean field,” Pacific Journal of Mathematics, vol. 50, pp. 169–185, 1974.
[23] A. A. Nasef and A. S. Farrag, “Completelyb-irresolute functions,” Proceedings of the Mathemat- ical and Physical Society of Egypt, no. 74, pp. 73–86, 1999.
Metin Akda˘g: Department of Mathematics, Faculty of Arts and Science, Cumhuriyet University, 58140 Sivas, Turkey
Email address:[email protected]
Special Issue on
Singular Boundary Value Problems for Ordinary Differential Equations
Call for Papers
The purpose of this special issue is to study singular boundary value problems arising in differential equations and dynamical systems. Survey articles dealing with interac- tions between different fields, applications, and approaches of boundary value problems and singular problems are welcome.
This Special Issue will focus on any type of singularities that appear in the study of boundary value problems. It includes:
• Theory and methods
• Mathematical Models
• Engineering applications
• Biological applications
• Medical Applications
• Finance applications
• Numerical and simulation applications
Before submission authors should carefully read over the journal’s Author Guidelines, which are located at http://www.hindawi.com/journals/bvp/guidelines.html. Au- thors should follow the Boundary Value Problems manu- script format described at the journal site http://www .hindawi.com/journals/bvp/. Articles published in this Spe- cial Issue shall be subject to a reduced Article Proc- essing Charge of C200 per article. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking Sys- tem at http://mts.hindawi.com/according to the following timetable:
Manuscript Due May 1, 2009 First Round of Reviews August 1, 2009 Publication Date November 1, 2009
Lead Guest Editor
Juan J. Nieto,Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de
Compostela, Santiago de Compostela 15782, Spain;
Guest Editor
Donal O’Regan,Department of Mathematics, National University of Ireland, Galway, Ireland;
Hindawi Publishing Corporation http://www.hindawi.com
