In 1997, Park et al. [5] offered a new notion called δ-semiopen sets which are stronger than semi-open sets but weaker than δ-open sets. It is the aim of this paper to introduce and study some properties of functions with strongly δ-semiclosed graphs by utilizing δ-semiopen sets and the δ-semi-closure operator

Caldas, M.

Jafari, S.

Noiri, T.

English

Università del Salento: ESE - Salento University Publishing

Note di Matematica 27, n. 1, 2007, 103–110.On functions with stronglyδ-semiclosed graphsM. CaldasDepartmento de Matematica Aplicada, Universidade Federal Fluminense,Rua Mario Santos Braga, s/n, 24020-140, Niteroi, RJ BRAZIL.gmamccs@vm.uff.brS. JafariCollege of Vestsjaelland South,Herrestraede 11, 4200 Slagelse, DENMARKjafari@stofanet.dkT. NoiriDepartment of Mathematics, Yatsushiro College of Technology,Yatsushiro, Kumamoto, 866-8501 JAPANnoiri@as.yatsushiro-nct.ac.jpReceived: 19/09/2005; accepted: 17/06/2006.Abstract. In 1997, Park et al. [5] offered a new notion called δ-semiopen sets which arestronger than semi-open sets but weaker than δ-open sets. It is the aim of this paper tointroduce and study some properties of functions with strongly δ-semiclosed graphs by utilizingδ-semiopen sets and the δ-semi-closure operator.Keywords: δ-semiopen set, δ-semiT1 space, δ-semiT2 space, strongly δ-semiclosed graph.MSC 2000 classification: primary 54B05, secondary 54C081 Introduction and preliminariesIn what follows (X, τ) and (Y, σ) (or X and Y ) denote topological spaces.Let A be a subset of X. We denote the interior, the closure and the complementof a set A by Int(A), Cl(A) and X\A or Ac, respectively.Levine [3] defined semiopen sets which are weaker than open sets in topolog-ical spaces. After Levine’s semiopen sets, mathematicians gave in several papersdifferent and interesting new modifications of open sets as well as generalizedopen sets. In 1968, Velic˘ko [6] introduced δ-open sets, which are stronger thanopen sets, in order to investigate the characterization of H-closed spaces. In1997, Park et al. [5] introduced the notion of δ-semiopen sets which are strongerthan semiopen sets but weaker than δ-open sets and investigated the relation-ships between several types of these open sets.A subset A of a topological space X is said to be δ-semiopen [5] (resp.104 M. Caldas, S. Jafari, T. Noirisemiopen [3]) set if there exists a δ-open (resp. open) set U of X such thatU ⊂ A ⊂ Cl(U). The complement of a δ-semiopen (resp. semiopen) set is calleda δ-semiclosed (resp. semiclosed) set. A point x ∈ X is called the δ-semiclusterpoint of A if A ∩ U 6= ∅ for every δ-semiopen set U of X containing x. Theset of all δ-semicluster points of A is called the δ-semiclosure of A, denoted byδClS(A). We denote the collection of all δ-semiopen (resp. δ-semiclosed) setsby δSO(X) (resp. δSC(X)). We set δSO(X,x) = {U : x ∈ U ∈ δSO(X)} andδSC(X,x) = {U : x ∈ U ∈ δSC(X)}.1 Lemma. (Park et al. [5]) The intersection (resp. union) of arbitrary col-lection of δ-semiclosed (resp. δ-semiopen) sets in (X, τ) is δ-semiclosed (resp.δ-semiopen).2 Corollary. Let A be a subset of a topological space (X, τ), then δClS(A)= ∩{F ∈ δSC(X, τ) : A ⊂ F}.3 Lemma. (Park et al. [5]) Let A, B and Ai (i ∈ I) be subsets of a space(X, τ), the following properties hold:(1) A ⊂ δClS(A).(2) If A ⊂ B, then δClS(A) ⊂ δClS(B).(3) δClS(A) is δ-semiclosed.(4) δClS(δClS(A)) = δClS(A).(5) A is δ-semiclosed if and only A = δClS(A).4 Corollary. (Caldas et al. [1]) Let Ai (i ∈ I) be subsets of a space (X, τ),the following properties hold:(1) δClS(∩{Ai : i ∈ I}) ⊂ ∩{δClS(Ai) : i ∈ I}.(2) δClS(∪{Ai : i ∈ I}) ⊃ ∪{δClS(Ai) : i ∈ I}.5 Definition. A topological space (X, τ) is called:(1) δ-semi T1 [1] if for any distinct pair of points x and y in X, there is a δ-semiopen U inX containing x but not y and δ-semiopen V inX containingy but not x,(2) δ-semi T2 [1] if for any distinct pair of points x and y in X, there existU ∈ δSO(X,x) and V ∈ δSO(X, y) such that U ∩ V = ∅.Recall that a function f : X → Y is said to be:On functions with strongly δ-semiclosed graphs 105(1) δ-semicontinuous [1] if for each x ∈ X and each V ∈ δSO(Y, f(x)), thereexists U ∈ δSO(X,x) such that f(U) ⊂ V , equivalently if the inverseimage of each δ-semiopen set is δ-semiopen,(2) quasi δ-semicontinuous [1] if for each x ∈ X and each V ∈ δSO(Y, f(x)),there exists U ∈ δSO(X,x) such that f(U) ⊂ δClS(V ).2 Strongly δ-semiclosed graphsLet f : (X, τ)→ (Y, σ) be any function, then the subset G(f) = {(x, f(x)) :x ∈ X} of the product space (X × Y, τ × σ) is called the graph of f [2].6 Definition. A function f : X → Y has a strongly δ-semiclosed graph iffor each (x, y) ∈ (X × Y )\G(f), there exist U ∈ δSO(X,x) and V ∈ δSO(Y, y)such that [U × δClS(V )] ∩G(f) = ∅.7 Lemma. A function f : X → Y , has a strongly δ-semiclosed graph if foreach (x, y) ∈ (X × Y )\G(f), there exist U ∈ δSO(X,x) and V ∈ δSO(Y, y)such that f(U) ∩ δClS(V ) = ∅.8 Theorem. If f : X → Y is a function with strongly δ-semiclosed graph,then for each x ∈ X, f(x) = ∩{δClS(f(U)) : U ∈ δSO(X,x)}.Proof. Suppose the theorem is false. Then there exists a point y ∈ Y suchthat y 6= f(x) and y ∈ ∩{δClS(f(U)) : U ∈ δSO(X,x)}. This implies that y ∈δClS(f(U)) for every U ∈ δSO(X,x). So V ∩f(U) 6= ∅ for every V ∈ δSO(Y, y).This, in its turn, indicates that δClS(V )∩f(U) ⊃ V ∩f(U) 6= ∅ which contradictsthe hypothesis that f is a function with a strongly δ-semiclosed graph. Hencethe theorem holds. QED9 Theorem. If f : X → Y is δ-semicontinuous and Y is δ-semi T2, thenG(f) is strongly δ-semiclosed.Proof. Let (x, y) ∈ (X × Y )\G(f). The δ-semi T2-ness of Y gives theexistence of a set V ∈ δSO(Y, y) such that f(x) /∈ δClS(V ).Now Y \δClS(V ) ∈ δSO(Y, f(x)). Therefore, by the δ-semicontinuity of f thereexists U ∈ δSO(X,x) such that f(U) ⊂ Y \δClS(V ). Consequently, f(U) ∩δClS(V ) = ∅ and therefore G(f) is strongly δ-semiclosed. QED10 Theorem. If f : X → Y is surjective and has a strongly δ-semiclosedgraph G(f), then Y is both δ-semi T2 and δ-semi T1.Proof. Let y1, y2 (y1 6= y2) ∈ Y . The surjectivity of f gives a x1 ∈ X suchthat f(x1) = y1. Now (x1, y2) ∈ (X × Y )\G(f). The strong δ-semiclosedness ofG(f) provides U ∈ δSO(X,x1) and V ∈ δSO(Y, y2) such that f(U)∩δClS(V ) =∅, whence one infers that y1 /∈ δClS(V ). This means that there exists W ∈106 M. Caldas, S. Jafari, T. NoiriδSO(Y, y1) such that W ∩ V = ∅. So, Y is δ-semi T2 and δ-semi T2-ness alwaysguarantees δ-semi T1-ness. Hence Y is δ-semi T1. QED11 Theorem. A space X is δ-semi T2 if and only if the identity functionid : X → X has a strongly δ-semiclosed graph G(id).Proof. Necessity. Let Y is δ-semi T2. Since the identity function id :X → X is δ-semicontinuous, it follows from Theorem 9 that G(id) is stronglyδ-semiclosed.Sufficiency. Let G(id) be strong δ-semiclosed graph. Then the surjectivityof id and strongly δ-semiclosedness of G(id) together imply, by Theorem 10, thatX is δ-semi T2. QED12 Theorem. If f : X → Y is an injection and G(f) is strongly δ-semiclosed, then X is δ-semi T1.Proof. Since f is injective, for any pair of distinct points x1, x2 ∈ X,f( x1) 6= f(x2). Then (x1, f(x2)) ∈ (X × Y )\G(f). Since G(f) is stronglyδ-semiclosed, there exist U ∈ δSO(X,x1) and V ∈ δSO(Y, f(x2)) such thatf(U)∩δClS(V ) = ∅. Therefore x2 /∈ U . Pursuing the same reasoning as before weobtain a set W ∈ δSO(X,x2) such that x1 /∈W . Hence Y is δ-semi T1. QED13 Theorem. If f : X → Y is a bijective function with a strongly δ-semiclosed graph, then both X and Y are δ-semi T1.Proof. The proof is an immediate consequence of Theorems 10 and 12.QED14 Theorem. If a quasi δ-semicontinuous function f : X → Y is an injec-tion with a strongly δ-semiclosed graph G(f), then X is δ-semi T2.Proof. Since f is injective, for any pair of distinct points x1, x2 ∈ X,f(x1) 6= f(x2). Therefore (x1, f(x2)) ∈ (X × Y )\G(f). Since G(f) is stronglyδ-semiclosed, there exist U ∈ δSO(X,x1) and V ∈ δSO(Y, f(x2)) such thatf(U) ∩ δClS(V ) = ∅, hence one obtains U ∩ f−1(δClS(V )) = ∅. Consequently,f−1(δClS(V )) ⊂ X\U . Since f is quasi δ-semicontinuous, it is so at x2. Thenthere exists W ∈ δSO(X,x2) such that f(W ) ⊂ δClS(V ). From this and theforegoing follow thatW ⊂ f−1(δClS(V )) ⊂ X\U , hence one infers thatW∩U =∅. Thus for the pair of distinct points x1, x2 ∈ X, there exist U ∈ δSO(X,x1)and W ∈ δSO(X,x2) such that W ∩U = ∅. This guarantees the δ-semi T2-nessof X. QED15 Corollary. If a δ-semicontinuous function f : X → Y is an injectionwith a strongly δ-semiclosed graph, then X is δ-semi T2.Proof. The proof follows from Theorem 14 and the fact that every δ-semicontinuous is quasi δ-semicontinuous. QEDOn functions with strongly δ-semiclosed graphs 10716 Theorem. If f : X → Y is quasi δ-semicontinuous bijective withstrongly δ-semiclosed graph, then both X and Y are δ-semi T2.Proof. The proof follows from Theorem 14 and 10. QEDFor the rest of this article we shall assume the Property △: δSO(X) closedunder finite intersections.17 Definition. X is called strongly △-closed (resp. A subset A of X is saidto be strongly △-closed relative to X), if every δ-semiopen cover of X (resp. ifevery cover of A by δ-semiopen sets) has a finite subfamily such that the unionof their δ-semiclosures covers X (resp. has a finite subfamily such that the unionof their δ-semiclosures covers X).18 Lemma. Every δ-semiclopen subset of a strongly △-closed space X isstrongly △-closed relative to X.Proof. Let B be any δ-semiclopen subset of a strongly △-closed spaceX. Let {Oλ : λ ∈ Ω} be any cover of B by δ-semiopen sets in X. Then thefamily F = {Oλ : λ ∈ Ω} ∪ {X\B} is a cover of X by δ-semiopen sets inX. Because of strongly △-closedness of X there exists a finite subfamily F ∗ ={Oλi : 1 ≤ i ≤ n} ∪ {X\B} of F such that the union of their δ-semiclosurescovers X. So, because of δ-semiclopenness of B we now infer that the family{δClS(Oλi) : 1 ≤ i ≤ n} covers B and hence B is strongly △-closed relative toX. QED19 Lemma. The δ-semiclosure of every δ-semiopen set is δ-semiopen.Proof. Every regular open set is δ-open and every δ-open set is δ-semiopen.Thus, every regular closed set is δ-semiclosed. Now let A be any δ-semiopenset. There exists a δ-open set U such that U ⊂ A ⊂ Cl(U). Hence, we haveU ⊂ δClS(U) ⊂ δClS(A) ⊂ δClS(Cl(U)) = Cl(U) since Cl(U) is regular closed.Therefore, δClS(A) is δ-semiopen. QED20 Theorem. Let f : X → Y be a function. If Y is a strongly △-closed,δ-semi-T2 space and G(f) is strongly δ-semiclosed, then f is quasi-δ-semicon-tinuous.Proof. Let x ∈ X and V ∈ δSO(Y, f(x)). Take any y ∈ Y \δClS(V ).Then (x, y) ∈ (X × Y )\G(f). Now the strong δ-semiclosedness of G(f) inducesthe existence of Uy(x) ∈ δSO(X,x) and Vy ∈ δSO(Y, y) such that f(Uy(x)) ∩δClS(Vy) = ∅. . . . (*).The δ-semi-T2-ness of Y implies the existence of Vy ∈ δSO(Y, y) such thatf(x) /∈ δClS(Vy). Now by Lemma 19 induces the δ-semiclopenness of δClS(V )and hence Y \δClS(V ) is also δ-semiclopen. Now {Vy : y ∈ Y \δClS(V )} is acover of Y \δClS(V ) by δ-semiopen sets in Y . By Lemma 18, there exists a finitesubfamily {Vyi : 1 ≤ i ≤ n} such that Y \δClS(V ) ⊂n⋃i=1δClS(Vyi). Let W =108 M. Caldas, S. Jafari, T. Noirin⋂i=1Uyi(x), where Uyi(x) are δ-semiopen sets in X satisfying (*). Since X enjoysthe Property △, W ∈ δSO(X,x). Now f(W ) ∩ (Y \δClS(V )) ⊂ f [n⋂i=1Uyi(x)] ∩(n⋃i=1δClS(Vyi)) =n⋃i=1(f [Uyi(x)] ∩ δClS(Vyi)) = ∅, by (*). Therefore, f(W ) ⊂δClS(V ) and this indicates that f is quasi-δ-semicontinuous. QED21 Corollary. If Y is a strongly △-closed then a surjerction f : X → Ywith a strongly δ-semiclosed graph is quasi-δ-semicontinuous.Proof. The proof follows from Theorem 10 and 19. QEDNoiri [4] showed that if G(f) is strongly closed then f has the followingproperty:(P ) For every set B quasi H–closed relative to Y , f−1(B) is a closed set of X.Analogously, we have22 Theorem. A f : X → Y has a strongly δ-semiclosed graph G(f), thenf enjoys the following property:(P ∗) For every set F which is strongly △-closed relative to Y , f−1(F ) isδ-semiclosed in X.Proof. If possible let f−1(F ) be not δ-semiclosed in X. Then there existsx ∈ δClS(f−1(F ))\ f−1(F ). Let y ∈ F . Then (x, y) ∈ (X × Y )\G(f). Stronglyδ-semiclosedness of G(f) gives the existence of Uy(x) ∈ δSO(X,x) and Vy ∈δSO(Y, y) such that f(Uy(x)) ∩ δClS(Vy) = ∅. . . .(*).Clearly {Vy : y ∈ F} is a cover of F by δ-semiopen sets in Y . The strong△-closedness of F relative to Y guarantees the existence of δ-semiopen setsVy1 , Vy2 , . . . , Vyn in Y such that F ⊂n⋃i=1δClS(Vyi).Let U =n⋂i=1Uyi(x), where Uyi(x) are the δ-semiopen sets in X satisfying (*).Since X enjoys the Property △, U ∈ δSO(X,x).Now f(U)∩F ⊂ f [n⋂i=1Uyi(x)]∩(n⋃i=1δClS(Vyi)) =n⋃i=1(f [Uyi(x)]∩δClS(Vyi)) = ∅.But x ∈ δClS(f−1(F )). Therefore U ∩f−1(F ) 6= ∅ which contradicts to theabove deduction. Hence the result is true. QED3 Additional properties23 Theorem. A space X is δ-semi T2 if and only if for every pair of pointsx, y ∈ X, x 6= y there exist U ∈ δSO(X,x), V ∈ δSO(X, y) such that δClS(U)∩δClS(V ) = ∅.On functions with strongly δ-semiclosed graphs 109Proof. Necessity. Suppose that X is δ-semi T2. Let x and y be distinctpoints of X. There exist δ-semiopen sets Ux and Uy such that x ∈ Ux, y ∈ Uyand Ux ∩ Uy = ∅. Hence δClS(Ux) ∩ δClS(Uy) = ∅ and by Lemma 19 δClS(Ux)is δ-semiopen. Therefore, we obtain δClS(Ux) ∩ δClS(Uy) = ∅.Sufficiency. This is obvious. QED24 Theorem. If Y is δ-semi T2 and f : X → Y is a quasi δ-semicontinuousinjection, then X is δ-semi T2.Proof. Since f is injective, for any pair of distinct points x1, x2 ∈ X,f(x1) 6= f(x2). By Theorem 23, the δ-semi T2 property of Y indicates thatthere exist Vi ∈ δSO(Y, f(xi)), i = 1, 2 such that δClS(V1) ∩ δClS(V2) = ∅.Hence f−1(δClS(V1)) ∩ f−1(δClS(V2)) = ∅. Since f is quasi δ-semicontinuous,there exists Ui ∈ δSO(X,xi), i = 1, 2 such that f(Ui) ⊂ δClS(Vi), i = 1, 2. It,then follows that Ui ⊂ f−1(δClS(Vi)), i = 1, 2.Hence U1∩U2 ⊂ f−1(δClS(V1))∩f−1(δClS(V2)) = ∅. This guarantees the δ-semiT2-ness of X. QED25 Definition. A function f : X → Y is called δ-semiopen if f(A) ∈δSO(Y ) for all A ∈ δSO(X).26 Theorem. If a bijection f : X → Y is δ-semiopen and X is δ-semi T2,then G(f) is strongly δ-semiclosed.Proof. Let (x, y) ∈ (X × Y )\G(f). Then y 6= f(x). Since f is bijective,x 6= f−1(y). SinceX is δ-semi T2, there exist Ux, Uy ∈ δSO(X) such that x ∈ Ux,f−1(y) ∈ Uy and Ux∩Uy = ∅. Moreover f is δ-semiopen and bijective, thereforef(x) ∈ f(Ux) ∈ δSO(Y ), y ∈ f(Uy) ∈ δSO(Y ) and f(Ux) ∩ f(Uy) = ∅. Hencef(Ux)∩δClS(f(Uy)) = ∅. This shows that G(f) is strongly δ-semiclosed. QED27 Theorem. If f : X → Y is quasi δ-semicontinuous and Y is δ-semi T2,then G(f) is strongly δ-semiclosed.Proof. Let (x, y) ∈ (X × Y )\G(f). Then y 6= f(x). Since Y is δ-semiT2, by Theorem 23 there exist V ∈ δSO(Y, y) and W ∈ δSO(Y, f(x)) suchthat δClS(V ) ∩ δClS(W ) = ∅. Since f is quasi δ-semicontinuous, there existsU ∈ δSO(X,x) such that f(U) ⊂ δClS(W ). This, therefore, implies that f(U)∩δClS(V ) = ∅. So by Lemma 7, G(f) is strongly δ-semiclosed. QEDWe close with the following questions which we were unable to answer:Question 1. Is there any nontrivial example of a function with a stronglyδ-semiclosed graph?Question 2. Are there nontrivial examples of the notions introduced inDefinition 17?Question 3. Is there a nontrivial example of a δ-semiopen function?110 M. Caldas, S. Jafari, T. NoiriAcknowledgements. The authors thank the referee for his remarks con-cerning this paper.References[1] M. Caldas, D. N. Georgiou, S. Jafari and T. Noiri: More on δ-semiopen sets, Notedi Matematica, 22(2) (2003/2004), 113–126.[2] T. Husain: Topology and Maps, Plenum press, New York, 1977.[3] N. Levine: Semi-open sets and semi-continuity in topological spaces, Amer. Math.Monthly, 70, (1963), 36–41.[4] T. Noiri: On functions with strongly closed graphs, Acta Math., Acad. Sci. Hungar., 32,(1978), 373–375.[5] J. H. Park, B. Y. Lee andM. J. Son: On δ-semiopen sets in topological space, J. IndianAcad. Math, 19, (1997), 1–4.[6] N. V. Velic˘ko: H-closed topological spaces, Mat. Sb., 70, (1966), 98–112; English transl.,in Amer. Math. Soc. Transl., (2),78, (1968), 103–118.

On functions with strongly δ-semiclosed graphs

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On functions with strongly δ-semiclosed graphs

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