A note on some applications of α open sets

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PII. S0161171203203161 http://ijmms.hindawi.com © Hindawi Publishing Corp.

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 of}_τ_⊂_τα_and_τα_{is 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);

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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);

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α

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);

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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 every}_V _∈_σ _{and, equivalently, if for each}_x_∈_X _and

each open setV ofY containingf (x), there existsU∈τα _with_x_∈_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 _containing_x_{. Hence, we have} _U_∩_(X_\_f−1_{(V ))}_≠_∅_for everyU∈τα _containing_x_{. It follows that}_x_∈_Cl

α(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 an}_α_{-open set of}_X_{. Thus,}_x_∈_Int

α(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);

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α

(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_}_If_A_{= {}_a_}_and_B_{= {}_b_}_{. Then, Ext}

α(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∈τα_containing_x_{such that}_{f (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

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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_{, where}_W _{is an} _α_{-open set of} X. Suppose thatx∈f−1_{(V )}_{. This means that}_{f (x)}_∈_V _{and also}_x_∈_W_{. By} the fact thatf is weaklyα-continuous, there existsU∈τα_containing_x_such

thatf (U)⊂Cl(V ). Therefore,U⊂f−1₍_Cl_{(V ))}_{. We can take}_U_{to be a subset of} W. It follows thatx∈U⊂f−1₍_Cl_{(V ))}_∩_W₌_f−1_{(V )}_{and thus the claim follows.}

References

[1] A. S. Mashhour, I. A. Hasanein, and S. N. El-Deeb,α-continuous andα-open map-pings, Acta Math. Hungar.41(1983), no. 3-4, 213–218.

[2] O. Nj˙astad,On some classes of nearly open sets, Paciﬁc J. Math.15(1965), 961– 970.

[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}