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A NOTE ON SOME APPLICATIONS OF α-OPEN SETS
MIGUEL CALDAS Received 16 March 2002
The object of this note is to introduce and study topological properties of α- derived,α-border,α-frontier, andα-exterior of a set using the concept ofα-open sets. Moreover, we study some further properties of the well-known notions of α-closure andα-interior. We also obtain a new decomposition ofα-continuous functions.
2000 Mathematics Subject Classification: 54C08, 54A05.
1. Introduction. The notion ofα-open set (originally calledα-sets) in topo- logical spaces was introduced by Nj˙astad [2] in 1965. Since then, it has been widely investigated in the literature. For these sets, we introduce the notions ofα-derived,α-border,α-frontier, andα-exterior of a set and show that some of their properties are analogous to those for open sets. Also, we give some additional properties ofα-closure andα-interior of a set due to Nj˙astad [2].
Throughout this paper,(X,τ)(simplyX) always mean topological spaces.
A subsetAof(X,τ)is calledα-open [2] ifA⊂Int(Cl(Int(A))). The comple- ment of anα-open set is calledα-closed. The intersection of allα-closed sets containingA is called theα-closure ofA, denoted by Clα(A). A subsetA is alsoα-closed if and only ifA=Clα(A). We denote the family ofα-open sets of(X,τ)byτα. It is shown in [2] (see also [4]) that each ofτ⊂ταandταis a topology onX.
2. Applications ofα-open sets
Definition2.1. LetAbe a subset of a spaceX. A pointx∈Ais said to be α-limit point ofAif for eachα-open setUcontainingx,U∩(A\{x})≠∅. The set of allα-limit points ofAis called anα-derived set ofAand is denoted by Dα(A).
Theorem2.2. For subsetsA,Bof a spaceX, the following statements hold:
(1) Dα(A)⊂D(A), whereD(A)is the derived set ofA; (2) ifA⊂B, thenDα(A)⊂Dα(B);
(3) Dα(A)∪Dα(B)⊂Dα(A∪B)andDα(A∩B)⊂Dα(A)∩Dα(B); (4) Dα(Dα(A))\A⊂Dα(A);
(5) Dα(A∪Dα(A))⊂A∪Dα(A).
Proof. (1) It suffices to observe that every open set isα-open.
(3) Follows by (2).
(4) If x ∈Dα(Dα(A))\A and U is an α-open set containingx, then U∩ (Dα(A)\{x})≠∅. Lety∈U∩(Dα(A)\{x}). Then, sincey∈Dα(A)andy∈ U,U∩(A\{y})≠∅. Letz∈U∩(A\{y}). Then, z≠x forz∈Aand x∉A. Hence,U∩(A\{x})≠∅. Therefore,x∈Dα(A).
(5) Letx∈Dα(A∪Dα(A)). Ifx∈A, the result is obvious. So, letx∈Dα(A∪ Dα(A))\A, then, forα-open set U containing x, U∩(A∪Dα(A)\{x})≠∅. Thus, U∩(A\{x})≠∅ or U∩(Dα(A)\{x})≠∅. Now, it follows similarly from (4) that U∩(A\{x})≠∅. Hence, x∈Dα(A). Therefore, in any case, Dα(A∪Dα(A))⊂A∪Dα(A).
In general, the converse of (1) may not be true and the equality does not hold in (3) ofTheorem 2.2.
Example2.3. LetX= {a,b,c}with topologyτ= {∅,{a},X}. Thus,τα= {∅,{a},{a,b},{a,c},X}. Take the following:
(i) A= {c}. Then,D(A)= {b}andDα(A)= ∅. Hence,D(A)⊆Dα(A); (ii) C= {a}andE= {b,c}. Then,Dα(C)= {b,c}andDα(E)= ∅. Hence,
Dα(C∪E)≠Dα(C)∪Dα(E).
Theorem2.4. For any subsetAof a spaceX,Clα(A)=A∪Dα(A).
Proof. SinceDα(A)⊂Clα(A),A∪Dα(A)⊂Clα(A). On the other hand, let x∈Clα(A). If x∈A, then the proof is complete. Ifx∉A, eachα-open set U containingxintersects Aat a point distinct fromx; sox∈Dα(A). Thus, Clα(A)⊂A∪Dα(A), which completes the proof.
Corollary2.5. A subsetAisα-closed if and only if it contains the set of its α-limit points.
Definition2.6. A pointx∈Xis said to be anα-interior point ofAif there exists anα-open setUcontainingxsuch thatU⊂A. The set of allα-interior points ofAis said to beα-interior ofA[1] and denoted by Intα(A).
Theorem2.7. For subsetsA,Bof a spaceX, the following statements are true:
(1) Intα(A)is the largestα-open set contained inA; (2) Aisα-open if and only ifA=Intα(A);
(3) Intα(Intα(A))=Intα(A); (4) Intα(A)=A\Dα(X\A); (5) X\Intα(A)=Clα(X\A); (6) X\Clα(A)=Intα(X\A); (7) A⊂B, thenIntα(A)⊂Intα(B); (8) Intα(A)∪Intα(B)⊂Intα(A∪B); (9) Intα(A)∩Intα(B)⊃Intα(A∩B).
α-OPEN SETS
Proof. (4) Ifx∈A\Dα(X\A), thenx∉Dα(X\A)and so there exists an α-open setU containingx such thatU∩(X\A)= ∅. Then, x∈U⊂A and hencex∈Intα(A), that is,A\Dα(X\A)⊂Intα(A). On the other hand, ifx∈ Intα(A), thenx∉Dα(X\A)since Intα(A)isα-open and Intα(A)∩(X\A)= ∅. Hence, Intα(A)=A\Dα(X\A).
(5)X\Intα(A)=X\(A\Dα(X\A))=(X\A)∪Dα(X\A)=Clα(X\A). Definition2.8. bα(A)=A\Intα(A)is said to be theα-border ofA. Theorem2.9. For a subsetAof a spaceX, the following statements hold:
(1) bα(A)⊂b(A)whereb(A)denotes the border ofA; (2) A=Intα(A)∪bα(A);
(3) Intα(A)∩bα(A)= ∅;
(4) Ais anα-open set if and only ifbα(A)= ∅; (5) bα(Intα(A))= ∅;
(6) Intα(bα(A))= ∅;
(7) bα(bα(A))=bα(A); (8) bα(A)=A∩Clα(X\A); (9) bα(A)=Dα(X\A).
Proof. (6) Ifx∈Intα(bα(A)), thenx∈bα(A). On the other hand, since bα(A)⊂A, x ∈Intα(bα(A))⊂ Intα(A). Hence, x ∈Intα(A)∩bα(A), which contradicts (3). Thus, Intα(bα(A))= ∅.
(8)bα(A)=A\Intα(A)=A\(X\Clα(X\A))=A∩Clα(X\A). (9)bα(A)=A\Intα(A)=A\(A\Dα(X\A))=Dα(X\A).
Example2.10. Consider the topological space(X,τ)given inExample 2.3.
IfA= {a,b}, thenbα(A)= ∅andb(A)= {b}. Hence,b(A)⊆bα(A), that is, in general, the converseTheorem 2.9(1) may not be true.
Definition2.11. Frα(A)=Clα(A)\Intα(A)is said to be theα-frontier ofA. Theorem2.12. For a subsetAof a spaceX, the following statements hold:
(1) Frα(A)⊂Fr(A)whereFr(A)denotes the frontier ofA; (2) Clα(A)=Intα(A)∪Frα(A);
(3) Intα(A)∩Frα(A)= ∅; (4) bα(A)⊂Frα(A);
(5) Frα(A)=bα(A)∪Dα(A);
(6) Ais anα-open set if and only if Frα(A)=Dα(A); (7) Frα(A)=Clα(A)∩Clα(X\A);
(8) Frα(A)=Frα(X\A); (9) Frα(A)isα-closed;
(10) Frα(Frα(A))⊂Frα(A); (11) Frα(Intα(A))⊂Frα(A); (12) Frα(Clα(A))⊂Frα(A); (13) Intα(A)=A\Frα(A).
Proof. (2) Intα(A)∪Frα(A)=Intα(A)∪(Clα(A)\Intα(A))=Clα(A). (3) Intα(A)∩Frα(A)=Intα(A)∩(Clα(A)\Intα(A))= ∅.
(5) Since Intα(A)∪Frα(A)= Intα(A)∪bα(A)∪Dα(A), Frα(A)=bα(A)∪ Dα(A).
(7) Frα(A)=Clα(A)\Intα(A)=Clα(A)∩Clα(X\A).
(9) Clα(Frα(A))=Clα(Clα(A)∩Clα(X\A))⊂Clα(Clα(A))∩Clα(Clα(X\A))= Frα(A).
Hence, Frα(A)isα-closed.
(10) Frα(Frα(A))=Clα(Frα(A))∩Clα(X\Frα(A))⊂Clα(Frα(A))=Frα(A). (12) Frα(Clα(A))= Clα(Clα(A))\Intα(Clα(A))= Clα((A))\Intα(Clα(A)) = Clα(A)\Intα(A)=Frα(A).
(13)A\Frα(A)=A\(Clα(A)\Intα(A))=Intα(A).
The converses of (1) and (4) of Theorem 2.12 are not true in general, as shown byExample 2.13.
Example2.13. Consider the topological space(X,τ)given inExample 2.3.
IfA= {c}, then Fr(A)= {b,c} ⊆ {c} =Frα(A), and ifB= {a,b}, then Frα(B)= {c} ⊆bα(B).
Definition2.14. A functionf:(X,τ)→(Y ,σ )is said to beα-continuous [1] if f−1(V )∈τα for everyV ∈σ and, equivalently, if for eachx∈X and each open setV ofY containingf (x), there existsU∈τα withx∈U such thatf (U)⊂V.
In the following theorem,α-c. denotes the set of pointsxofXfor which a functionf:(X,τ)→(Y ,σ )is notα-continuous.
Theorem 2.15. α-c. is identical with the union of theα-frontiers of the inverse images ofα-open sets containingf (x).
Proof. Suppose thatf is notα-continuous at a pointx ofX. Then, there exists an open setV⊂Y containing f (x)such thatf (U) is not a subset of V for everyU∈τα containingx. Hence, we have U∩(X\f−1(V ))≠∅for everyU∈τα containingx. It follows thatx∈Clα(X\f−1(V )). We also have x∈f−1(V )⊂Clα(f−1(V )). This means thatx∈Frα(f−1(V )).
Now, let f beα-continuous atx∈X and V⊂Y any open set containing f (x). Then,x∈f−1(V )is anα-open set ofX. Thus,x∈Intα(f−1(V )) and thereforex∉Frα(f−1(V ))for every open setVcontainingf (x).
Definition2.16. Extα(A)=Intα(X\A)is said to be anα-exterior ofA. Theorem2.17. For a subsetAof a spaceX, the following statements hold:
(1) Ext(A)⊂Extα(A)whereExt(A)denotes the exterior ofA; (2) Extα(A)isα-open;
(3) Extα(A)=Intα(X\A)=X\Clα(A); (4) Extα(Extα(A))=Intα(Clα(A));
α-OPEN SETS (5) IfA⊂B, thenExtα(A)⊃Extα(B);
(6) Extα(A∪B)⊂Extα(A)∪Extα(B); (7) Extα(A∩B)⊃Extα(A)∩Extα(B); (8) Extα(X)= ∅;
(9) Extα(∅)=X;
(10) Extα(A)=Extα(X\Extα(A)); (11) Intα(A)⊂Extα(Extα(A)); (12) X=Intα(A)∪Extα(A)∪Frα(A).
Proof. (4) Extα(Extα(A)) = Extα(X\Clα(A)) = Intα(X\(X\Clα(A))) = Intα(Clα(A)).
(10) Extα(X\Extα(A)) = Extα(X\Intα(X\A)) = Intα(X\(X\Intα(X\A))) = Intα(Intα(X\A))=Intα(X\A)=Extα(A).
(11) Intα(A) ⊂ Intα(Clα(A)) = Intα(X\Intα(X\A)) = Intα(X\Extα(A)) = Extα(Extα(A)).
Example2.18. LetX= {a,b,c,d}with topologyτ= {∅,{c,d},X}. Hence, τα= {∅,{c,d},{b,c,d},{a,c,d},X}IfA= {a}andB= {b}. Then, Extα(A)⊆ Ext(A), Extα(A∩B)≠Extα(A)∩Extα(B), and Extα(A∪B)≠Extα(A)∪Extα(B).
3. A new decomposition ofα-continuity
Definition 3.1. A function f : (X,τ)→ (Y ,σ ) is said to be weakly α- continuous [3] if, for eachx∈Xand each open setV ofY containingf (x), there existsU∈ταcontainingxsuch thatf (U)⊂Cl(V ).
Theorem3.2(Noiri [3]). A functionf:(X,τ)→(Y ,σ )is weaklyα-contin- uous if and only if, for every open setV of Y,f−1(V )⊂Intα(f−1(Cl(V ))).
The following notion is motivated by the above theorem.
Definition3.3. A functionf:(X,τ)→(Y ,σ )is relatively weaklyα-contin- uous, if for eachx ∈X and each open setV in Y containing f (x), the set f−1(V )isα-open in the subspacef−1(Cl(V )).
Theorem3.4. Anα-continuous function is relatively weaklyα-continuous.
Proof. Straightforward.
The following example shows that the converse ofTheorem 3.4is not true.
Example3.5. LetXbe the set of all real numbers,τthe indiscrete topology forX, andσ the discrete topology forX. Letf:(X,τ)→(X,σ )be the identity function. Then,f is relatively weaklyα-continuous but it is not weakly α- continuous (hence it is notα-continuous) because Intα(f−1(Cl(V )))= ∅for any subsetVof(X,σ ).
Example 3.6. Let X = {a,b,c}, τ = {∅,X,{c}}, and σ = {∅,X,{a},{b}, {a,b}}. Letf:(X,τ)→(X,σ )be the identity function. Then,f is weaklyα- continuous but not relatively weaklyα-continuous.
Examples3.5and3.6show that weaklyα-continuous and relatively weakly α-continuous are independent.
The significance of relatively weaklyα-continuous is that it yields a decom- position ofα-continuous with weaklyα-continuous as the other factor.
Theorem3.7. A functionf:(X,τ)→(Y ,σ )isα-continuous if and only if it is weaklyα-continuous and relatively weaklyα-continuous.
Proof. The necessity is given byTheorem 3.4and by the fact that every α-continuous function is weaklyα-continuous.
Sufficiency. LetV be an open set in Y. Sincef is relatively weaklyα- continuous, we havef−1(V )=f−1(Cl(V ))∩W, whereW is an α-open set of X. Suppose thatx∈f−1(V ). This means thatf (x)∈V and alsox∈W. By the fact thatf is weaklyα-continuous, there existsU∈ταcontainingxsuch thatf (U)⊂Cl(V ). Therefore,U⊂f−1(Cl(V )). We can takeUto be a subset of W. It follows thatx∈U⊂f−1(Cl(V ))∩W=f−1(V )and thus the claim follows.
References
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[2] O. Nj˙astad,On some classes of nearly open sets, Pacific J. Math.15(1965), 961–
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[3] T. Noiri,Weaklyα-continuous functions, Int. J. Math. Math. Sci.10(1987), no. 3, 483–490.
[4] T. Ohba and J. Umehara,A simple proof ofτα being a topology, Mem. Fac. Sci.
Kôchi Univ. Ser. A Math.21(2000), 87–88.
Miguel Caldas: Departamento de Matemática Aplicada, Universidade Federal Fluminense-IMUFF, Rua Mário Santos Braga s/n0, CEP:24020-140, Niteroi, R.J., Brasil
E-mail address:[email protected]
