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γ-SETS AND γ -CONTINUOUS FUNCTIONS
WON KEUN MIN Received 17 September 2001
We introduce a new class of sets, calledγ-sets, and the notion ofγ-continuity and investi- gate some properties and characterizations. In particular,γ-sets andγ-continuity are used to extend known results for semi-open sets and semi-continuity.
2000 Mathematics Subject Classification: 54C08, 54A10, 54A20.
1. Introduction. Let X, Y, and Z be topological spaces on which no separation axioms are assumed unless explicitly stated. LetSbe a subset ofX. The closure (resp., interior) ofSwill be denoted by clS(resp., intS). A subsetSofXis called a semi-open set [2] (resp.,α-set [4]) ifS⊂cl(int(S))(resp.,S⊂int(cl(int(S)))). The complement of a semi-open set (resp.,α-set) is called semi-closed set (resp.,α-closed set). The family of all semi-open sets (resp.,α-sets) inXwill be denoted by SO(X)(resp.,α(X)). A function f :X→Y is called semi-continuous [2] (resp., α-continuous [3]) if f−1(V )∈SO(X) (resp.,f−1(V )∈α(X)) for each open setVofY. A functionf:X→Y is called semi- open [2] (resp.,α-open [3]) if for every semi-open (resp.,α-open) setU inX,f (U)is semi-open (resp.,α-open) inY.
A subsetM(x)of a spaceXis called a semi-neighborhood of a pointx∈Xif there exists a semi-open setSsuch thatx∈S⊂M(x). In [1], Latif introduced the notion of semi-convergence of filters and investigated some characterizations related to semi- open continuous functions. Now, we recall the concept of semi-convergence of filters.
LetS(x)= {A∈SO(X):x∈A}and letSx= {A⊂X:∃µ⊂S(x)such thatµis finite and∩µ⊂A}. Then,Sxis called the semi-neighborhood filter atx. For any filterFonX, we say thatFsemi-converges toxif and only ifF is finer than the semi-neighborhood filter atx.
2. γ-sets
Definition2.1. Let(X,τ)be a topological space. A subsetUofXis called aγ-set inXif whenever a filterFsemi-converges toxandx∈U,U∈F.
The class of allγ-sets inX will be denoted byγ(X). In particular, the class of all γ-sets induced by the topologyτwill be denoted byγτ.
Remark 2.2. From the definition of semi-neighborhood filter andγ-set, we can easily say that every semi-open set is aγ-set, but the converse is always not true.
Example2.3. LetXbe the real number set with the usual topology. For eachx∈X, since both(a,x]and[x,b)are semi-open sets containingx, wherea < x < b,{x}is
an element ofSx. For any filterF onX, ifFsemi-converges toxand sinceF includes Sx, thenxis aγ-set. But it is not semi-open.
Remark2.4. In a topological space(X,τ), it is always true that
τ⊂α(X)⊂SO(X)⊂γ(X). (2.1) Theorem2.5. Let(X,τ)be a topological space. The intersection of finitely many semi-open subsets inXis aγ-set.
Proof. LetU1andU2be semi-open sets inX. For eachx∈U1∩U2, we getU1∩U2∈ Sx. Thus, from the concept of the semi-convergence of filters, whenever every filterF semi-converges tox,U1∩U2∈F.
Definition2.6. Let(X,τ)be a topological space. Theγ-interior of a setAinX, denoted by intγ(A), is the union of allγ-sets contained inA.
Theorem2.7. Let(X,τ)be a topological space andA⊂X. (a) intγ(A)= {x∈A:A∈Sx}.
(b)Aisγ-set if and only ifA=intγ(A).
Proof. (a) For eachx∈intγ(A), there exists aγ-setUsuch thatx∈UandU⊂A. From the notion ofγ-set, the subsetUis in the semi-neighborhood filterSx. SinceSxis a filter,A∈Sx. Conversely, letx∈AandA∈Sx, then there existU1···Un∈S(x)such thatU=U1∩···∩Un⊂A. ByTheorem 2.5,Uis aγ-set andU⊂A. Thusx∈intγ(A).
(b) The proof is obvious.
Theorem2.8. Let(X,τ)be a topological space. Then, the classγ(X)of allγ-subsets inXis a topology onX.
Proof. Since∅andX are semi-open, they are alsoγ-sets inX. LetA,B∈γ(X), x∈A∩B, and letFbe a filter. Suppose the filterFsemi-converges tox. ThenA,B∈F and sinceF is a filter,(A∩B)∈F. Thus,A∩Bis aγ-set.
For eachα∈IletAα∈γ(X)andU= ∪Aα. For eachx∈Uand for a filterF semi- converging toxthere exists a subsetAαofUsuch thatx∈Aα, and sinceAαisγ-set, it is obvious thatAα∈F. SinceF is a filter,U is an element of the filterF and thus U= ∪Aαis aγ-set.
In a topological space(X,τ), the class of allγ-sets induced by the topologyτwill be denoted by(X,γτ). A subsetBofXis called aγ-closed set if the complement ofB is aγ-set. Thus, the intersection of any family ofγ-closed sets is aγ-closed set and the union of finitely manyγ-closed sets is aγ-closed set.
Obviously, we obtain the following theorem by definition of theγ-set.
Theorem2.9. Let(X,τ)be a topological space. A setGis γ-closed if and only if wheneverF semi-converges toxandA∈F,x∈A.
Definition2.10. Let(X,τ)be a topological space andA⊂X, clγ(A)=
x∈X:A∩U≠∅ ∀U∈Sx
. (2.2)
We call clγ(A)theγ-closure of the setA.
Now we can get the following theorem.
Theorem2.11. Let(X,τ)be a topological space and letAbe a subset ofX. Then the following properties hold:
(1) A⊂clγ(A);
(2) Aisγ-closed if and only ifA=clγA; (3) intγ(A)=X−clγ(X−A);
(4) clγ(A)=X−intγ(X−A).
3. γ-continuous andγ-irresolute functions
Definition3.1. Let(X,τ)and(Y ,µ)be topological spaces. A functionf:X→Y is calledγ-continuous if the inverse image of each open set ofY is aγ-set inX.
Since the class of allγ-sets in a given topological space is also a topology, we get the following equivalent statements.
Theorem3.2. Let(X,τ)and(Y ,µ)be topological spaces. Iff:(X,τ)→(Y ,µ)is a function, then the following statements are equivalent:
(1) fisγ-continuous;
(2) the inverse image of each closed set inY isγ-closed;
(3) clγ(f−1(B))⊂f−1(cl(B))for everyB⊂Y; (4) f (clγ(A))⊂cl(f (A))for everyA⊂X;
(5) f−1(int(B))⊂intγ(f−1(B))for everyB⊂Y.
Theorem3.3. Letf:(X,τ)→(Y ,µ)be a function between topological spaces. Then the following statements are equivalent:
(1) fisγ-continuous atx;
(2) if a filterF semi-converges tox, thenf (F)converges tof (x);
(3) forx∈Xand for each neighborhoodUoff (x), there is a subsetV∈Sxsuch thatf (V )⊂U.
Proof. (1)⇒(2). LetV be any open neighborhood off (x)inY. Thenf−1(V )is a γ-set containingx. Thusf−1(V )is an element inSx. SinceFsemi-converges toxand f (F)is a filter,V∈f (F). Consequently,f (F)converges tof (x).
(2)⇒(3). LetUbe anyγ-neighborhood off (x). Since alwaysSxsemi-converges tox, from the hypothesisSf (x)⊂f (Sx), and soU∈f (Sx). Thus, there is a subsetV∈Sx
such thatf (V )⊂U.
(3)⇒(1). The proof is obvious.
We can easily verify the following result.
Corollary 3.4. Letf :(X,τ)→(Y ,µ) be a function. If f is semi-continuous at x∈X, then whenever a filterFsemi-converges toxinX,f (F)converges tof (x)inY. Remark3.5. The following example shows that the converse ofCorollary 3.4may not be true. And we say that everyγ-continuous function is semi-continuous.
Example3.6. LetRbe the set of real numbers with the usual topology. We define f :R→Rbyf (x)=0, ifx∈Qand otherwise,f (x)=√
2. Clearly, a filter F semi-
converges toxif and only if ˙x⊂F. Thus ˙f (x)⊂f (F)and sof (F)converges tof (x). For an open interval(−1,1)containing 0,f−1{(−1,1)} =Q. SinceQis not semi-open inR,fis not semi-continuous.
Definition3.7. Let(X,τ)and(Y ,µ)be topological spaces. A functionf:X→Y is calledγ-irresolute if the inverse image of eachγset ofY is aγ-set inX.
The following theorems are obtained byDefinition 3.7.
Theorem3.8. Letf:(X,τ)→(Y ,µ)be a function between topological spaces. Then the following statements are equivalent:
(1) fisγ-irresolute;
(2) the inverse image of eachγ-closed set inY is aγ-closed set;
(3) clγτ(f−1(V ))⊂f−1(clγµ(V ))for everyV⊂Y; (4) f (clγτ(U))⊂clγµ(f (U))for everyU⊂X;
(5) f−1(intγµ(B))⊂intγτ(f−1(B))for everyB⊂Y.
Theorem3.9. Letf:(X,τ)→(Y ,µ)be a function between topological spaces. Then the following statements are equivalent:
(1) fisγ-irresolute;
(2) for x ∈X and for each V ∈ Sf(x), there exists an element U in the semi- neighborhood filterSxsuch thatf (U)⊂V;
(3) for eachx∈X, if a filterF semi-converges tox, thenf (F) semi-converges to f (x)inY.
Proof. (1)⇒(2). The proof is obvious.
(2)⇒(3). LetVbe an element of the semi-neighborhood filter ofSf (x) andUbe an element ofSxand letFbe a filter onXsemi-converging tox. Thenf (Sx)⊂f (F). Since U is an element inSx andf (F)is a filter, we can say thatV∈f (F). Consequently, f (F)semi-converges tof (x).
(3)⇒(1). Let V be anyγ-set inY and supposef−1(V )is not empty. For eachx∈ f−1(V ), since the semi-neighborhood filterSxsemi-converges toxand the hypothesis, clearly,f (Sx)semi-converges tox. And sinceV isγ-set containingf (x)andSf (x)⊂ f (Sx),V ∈f (Sx). Now we can take someγ-setU in Sx such thatf (U)⊂V. Thus, U⊂f−1(V )and sinceSxis a filter, so f−1(V )is an element ofSx. Andf−1(V )is a γ-set inXfromTheorem 2.7(b).
Corollary3.10. Letf:(X,τ)→(Y ,µ)be a function. Iffis irresolute, then when- ever a filterFsemi-converges toxinX,f (F)semi-converges tof (x)inY.
Remark3.11. We can get the following diagrams:
continuity ⇒α-continuity ⇒ semi-continuity ⇒γ-continuity;
α-irresolute ⇒ irresolute ⇒γ-irresolute. (3.1) Definition 3.12. For two topological spaces (X,τ) and (Y ,µ), a function f : (X,τ)→(Y ,µ)isγ-open if for every open setGinX,f (G)is aγ-set inY.
Theorem 3.13. Let f :(X,τ)→(Y ,µ)be a function between topological spaces.
Then,fisγ-open if and only ifint(f−1(B))⊂f−1(intγµ(B)), for eachB⊂Y.
Proof. LetB⊂Y andx∈int(f−1(B)). Then,f (int(f−1(B)))is aγ-set containing f (x). Sincef (int(f−1(B)))∈Sf (x)andSf (x)is a filter,B∈Sf (x). Thus,f (x)∈intγµ(B) and sox∈f−1(intγµ(B)).
Conversely, letAbe an open inXandy∈f (A). Then, A⊂int
f−1f (A)
⊂f−1 intγµ
f (A)
. (3.2)
Letx∈Abe such thatf (x)=y, thenx∈f−1(intγµ(f (A))). Then,y∈intγµ(f (A)), and fromTheorem 2.7(b)f (A)is aγ-set.
Remark3.14. If any function is semi-open, then it is alsoγ-open. But the converse may not hold. Consider a functionf:R→Rdefined byf (x)=0 for allx∈R, where the real number setRwith the usual topology. Thenf isγ-open. For any semi-open setG,f (G)= {0}and{0}is not semi-open set, thusf is not semi-open.
Theorem3.15. Letf:(X,τ)→(Y ,µ)be a function between topological spaces. The functionf isγ-open if and only if for eachx∈Xand for each neighborhoodGofx, f (G)is also an element of semi-neighborhood filterSf (x)inY.
Proof. LetGbe a neighborhood ofx, then there exists an open setUsuch that x∈U⊂G. Sincef isγ-open,f (x)∈f (U)=intγµ(f (U)), and sof (U)∈Sf (x). Since Sf (x)is a filter,f (G)∈Sf (x).
Conversely, letB⊂Y and x∈int(f−1(B)), then since int(f−1(B))is an element ofSxandSxis a filter,f−1(B)∈Sx. By the hypothesisf (f−1(B))∈Sf (x), and since Sf (x)is a filter,Bis also an element ofSf (x). ByDefinition 2.6,f (x)∈intγµ(B)and by Theorem 3.13, the functionf isγ-open.
Remark3.16. Now we get the following diagram:
open function ⇒α-open function ⇒semi-open function ⇒γ-open function. (3.3)
References
[1] R. M. Latif,Semi-convergence of filters and nets, to appear in Soochow J. Math.
[2] N. Levine,Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly 70(1963), 36–41.
[3] A. S. Mashhour, I. A. Hasanein, and S. N. El-Deeb,α-continuous andα-open mappings, Acta Math. Hungar.41(1983), no. 3-4, 213–218.
[4] O. Nj˙astad,On some classes of nearly open sets, Pacific J. Math.15(1965), no. 3, 961–970.
Won Keun Min: Department of Mathematics, Kangwon National University, Chun- cheon200-701, Korea
E-mail address:[email protected]
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