CUBO A Mathemati al Journal Vol.15, N o 03, (51�58). O tober 2013 Generalization of New Continuous Fun tions in Topologi al Spa es P. G. Patil Department of Mathemati s, SKSVM Agadi College of Engg. and Te hn., Laxmeshwar-582 116, Karnataka State, India. pgpatil01�gmail. om T. D. Rayanagoudar Department of Mathemati s, Govt.First Grade College, Annigeri-582 116, Karnataka State, India. rgoudar1980�gmail. om S. S. Ben halli Department of Mathemati s, Karnatak University, Dharwad-580 003 Karnataka State, India. ben halliss�gmail. om ABSTRACT In this paper, ωα- losed sets and ωα-open sets are used to de�ne and investigate the new lasses of fun tions namely somewhat ωα- ontinuous fun tions and totally ωα- ontinuous fun tions. RESUMEN En este artí ulo onjuntos errados-ωα y abiertos-ωα se usan para de�nir e investigar las lases de nuevas fun iones ontinuas ωα y totalmente ontinuas ωα. Keywords and Phrases: ωα- losed, ωα-open, ωα- ontinuous, somewhat ωα - ontinuous and totally ωα- ontinuous fun tions. 2010 AMS Mathemati s Subje t Classi� ation: 54C08, 54C10. 52 P.G.Patil, T.D. Rayanagoudar & S.S.Ben halli CUBO 15, 3 (2013) 1 Introdu tion Re ent progress in study of hara treization and generalization of ontinuity has been done by means of several generalized losed sets. As a generalization of losed sets ωα- losed sets were introdu ed and studied by Ben halli.et.al[1℄. The on epts of feebly ontinuous fun tions and feebly open fun tions were introdu ed by Zdenek Frolik[2℄. Gentry and Hoyle[3℄ introdu ed and studied the on epts of somewhat on- tinuous fun tions and somewhat open fun tions. Re ently, Santhileela and Balasubramanian[8℄ introdu ed and studied the on epts of somewhat semi ontinuous fun tions and somewhat semi open fun tions. In this paper, we will ontinue the study of related fun tions with ωα- losed and ωα-open sets. We introdu e and hara terize the on ept of somewhat ωα- ontnuous and totally ωα- ontinuous fun tions. 2 Preliminaries Throughout this paper (X, τ), (Y, σ) and (Z, η)(or simply X,Y and Z) represent topologi al spa es on whi h no separation axioms are assumed unless otherwise mentioned.For a subset A of (X, τ), cl(A),int(A), αcl(A) and Ac denote the losure of A, inerior of A, the α- losure of A and the ompliment of A in X respe ively. We re all the following de�nitions, whi h are usefull in the sequel.Before entering into our work we re all the following de�nitions from various authors. De�nition 2.1. A subset A of a topologi al spa e (X, τ) is alled semi-open [5℄ (resp. α-open[6℄) if A ⊆ cl(Int(A)) (resp A ⊆ Int(cl(Int(A))).The ompliment of semi-open (resp.α-open) is alled semi- losed(resp.α- losed). De�nition 2.2. A subset A of a topologi al spa e (X, τ) is alled ωα- losed [1℄ if αcl(A) ⊆ U whenever A ⊂ U and U is ω-open in X. The ompliment of ωα- losed set is ωα-open. The family of all ωα- losed sets of X is denoted by τ∗ ωα . In [7℄, we showed that τ∗ ωα forms a topology on X. De�nition 2.3. A fun tion f : (X, τ) → (Y, σ) is is said to be ωα- ontinuous [7℄ if the inverse image of every open set in Y is ωα-open in X. De�nition 2.4. A fun tion f : (X, τ) → (Y, σ) is is said to be perfe tly ωα- ontinuous [7℄ if the inverse image of every ωα open set in Y is lopen in X. De�nition 2.5. A fun tion f : (X, τ) → (Y, σ) is is said to be somewhat- ontinuous [3℄(resp.somewhat semi- ontinuous[8℄) if for U ∈ σ and f−1(U) 6= φ there exists an open (resp.semi open) set V in X su h that V 6= φ and V ⊆ f−1(U). CUBO 15, 3 (2013) Generalization of New Continuous Fun tions in Topologi al Spa es 53 Remark 2.6. Every somewhat ontinuous fun tion is somewhat semi ontinuous but onverse need not true in general[8℄. De�nition 2.7. A fun tion f : (X, τ) → (Y, σ) is said to be somewhat-open [3℄(resp.somewhat semi-open[8℄) fun tion provided that for U ∈ τ and U 6= φ, there exists an open (resp.semi open) set V in Y su h that V 6= φ and V ⊆ f−1(U). Remark 2.8. Every somewhat open fun tion is somewhat semi open fun tion but the onverse need not be true in general[8℄. 3 Somewhat ωα - Continuous fun tions In this se tion, we introdu e a new lass of fun tions alled somewhat ωα- ontinuous fun tions using ωα- losed sets and obtain some of their hara terizations. De�nition 3.1. A fun tion f : (X, τ) → (Y, σ) is said to be Somewhat ωα- ontinuous if for every open set U in Y and f−1(U) 6= φ, there exists ωα-open set V in X su h that V 6= φ and V ⊆ f−1(U). Example 3.2. Let X = Y = {p, q}, τ = {X, φ, } and σ = {X, φ, {p}}. The identity fun tion f : (X, τ) → (Y, σ) is somewhat ωα- ontinuous fun tion. Theorem 3.3. Every somewhat ontinuous fun tion is somewhat ωα- ontinuous but onverse need not true in general. Example 3.4. In Example 3.2, f is somewhat ωα- ontinuous but not somewhat ontinuous. Remark 3.5. The on ept of somewhat ωα- ontinuous and somewhat semi- ontinuous fun tions are independet as seen from the following examples. Example 3.6. In Example 3.2,f is somewhat ωα- ontinuous but not somewhat-semi ontinuous. Example 3.7. Let X = Y = {a, b, c}, τ = {X, φ, {a, b}} and σ = {X, φ, {a}}. Then the identity map f : (X, τ) → (Y, σ) is somewhat-semi ontinuous but not somewhat ωα- ontinuous. Theorem 3.8. If f : (X, τ) → (Y, σ) is somewhat ωα- ontinuous and g : (Y, σ) → (Z, η) is ontinuous fun tion,then their omposition gof is somewhat ωα- ontinuous fun tion. Proof. Let U be an open set in Z.Suppose that f−1(U) 6= φ. Sin e U is open and g is ontinuous, g−1(U) ∈ η. Suppose that f−1(g−1(U)) 6= φ. By hypothesis, there exists a ωα-open set V in Y su h that V 6= φ and V ⊆ f−1(g−1(U)) = (gof)−1(V). Therefore gof is somewhat ωα- ontinuous fun tion. Remark 3.9. In the above Theorem 3.8, if f is ontinuous and g is somewhat ωα- ontinuous then their omposition gof need not be somewhat ωα- ontinuous fun tion as seen from the following example. 54 P.G.Patil, T.D. Rayanagoudar & S.S.Ben halli CUBO 15, 3 (2013) Example 3.10. Let X = Y = Z = {p, q}, τ = {X, φ, {p}} , σ = {Y, φ, {p}} and η = {Z, φ, {q}} De�ne the fun tions f : (X, τ) → (Y, σ) by f(p) = f(q) = q and g : (Y, σ) → (Z, η) by g(p) = q and g(q) = p.Then learly f is ontinuous fun tion and g is somewhat ωα- ontinuous fun tion but their omoposition gof : (X, τ) → (Z, η) is not somewhat ωα- ontinuous fun tion. De�nition 3.11. A subset M of a topologi al spa e X is said to be ωα-dense in X if there is no proper ωα- losed set F in X su h that M ⊂ F ⊂ X. Theorem 3.12. The following statements are equivalent for a fun tion f : (X, τ) → (Y, σ): (1) f is somewhat ωα- ontinuous fun tion (2) If F is a losed subset of Y su h that f−1(F) 6= X,then there is a proper ωα- losed subset D of X su h that f−1(F) ⊂ D. (3) If M is a ωα-dense subset of X, then f(M) is a dense subset of Y. Proof. (1) ⇒ (2): Let F be a losed subset of Y su h that f−1(F) 6= X.Then f−1(Y−F) = X−f−1(f) 6= φ. Then from (1) there exists ωα-open set V in X su h that V 6= φ and V ⊂ f−1(Y − F) = X − f−1(F).This implies f−1(F) ⊂ X − V and X − V = D is a ωα- losed set inX. (2) ⇒ (3): Let M be any ωα-dense set in X. Suppose f(M) is not a dense subset of Y, then there exists a proper losed set F in Y su h that f(M) ⊂ F ⊂ Y. This implies f−1(F) 6= X. Then from (2) there exists a proper ωα- losed set D su h that M ⊂ f−1(F) ⊂ D ⊂ X. This ontradi ts the fa t that M is a ωα-dense set in X. (3) ⇒ (2): Suppose (2) is not true.Then there exists a losed setF in Y su h that f−1(F) 6= X.But there is no proper ωα- losed set D in X su h that f−1(F) ⊆ D. This means that f−1(F) is ωα-dense in X. But from hypothesis f(f−1(F)) = F must be dense in Y, whi h is ontradi tion to the hoi e of F. (2) ⇒ (1):Let U be an open set in Y and f−1(U) 6= φ. Then f−1(Y − U) = X − f−1(U) = φ. Then by hypothesis, there exists a proper ωα- losed set D su h that f−1(Y − U) ⊂ D. This implies that X − D ⊂ f−1(U) and X − D is ωα-open and X − D 6= φ. Theorem 3.13. Let f : (X, τ) → (Y, σ) be a fun tion and X = A ∪ B, A and B are open subsets of X su h that (f/A) and (f/B) are somewhat ωα- ontinuous fun tions then f is somewhat ωα- ontinuous fun tion. Proof. Let U be an open set in Y su h thatf−1(U) 6= φ. Then (f/A)−1(U) 6= φ or (f/B)−1(U) 6= φ or both (f/A)−1(U) 6= φ and (f/B)−1(U) 6= φ . ase(i): Suppose (f/A)−1(U) 6= φ. Sin e f/A is somewhat ωα- ontinuous , then there exists ωα open set V in A su h that V 6= φ and V ⊂ (f/A)−1(U) ⊂ f−1(U). Sin e V is ωα-open in A and A is open in X, V is ωα-open X . Hen e f is somewhat ωα- ontinuous fun tion. ase(ii): Suppose (f/B)−1(U) 6= φ. Sin e f/B is somewhat ωα- ontinuous , then there exists ωα open set V in B su h that V 6= φ and V ⊂ (f/B)−1(U) ⊂ f−1(U). Sin e V is ωα-open in B and B CUBO 15, 3 (2013) Generalization of New Continuous Fun tions in Topologi al Spa es 55 is open in X, V is ωα-open X . Hen e f is somewhat ωα- ontinuous fun tion. ase(iii): Suppose (f/A)−1(U) 6= φ and (f/B)−1(U) 6= φ. Follows from ase(i) and ase(ii). Theorem 3.14. If A be any set in X and f : (X, τ) → (Y, σ) be somewhat ωα- ontinuous su h that f(A) is dense in Y. Then any extension F of f is somewhat ωα- ontinuous. Proof. Let U be an open set inY su h thatF−1(U) 6= φ. Sin e f(A) ⊂ Y is dense in Y and U ∩ f(A) 6= φ. It follows that F−1(U) ∩ A 6= φ. That is f−1(U) ∩ A 6= φ.Hen e by hypothesis there exists a ωα-open set V in A su h that V 6= φ and V ⊂ f−1(U) ⊂ F−1(U).This implies F is somewhat ωα- ontinuous. De�nition 3.15. A topologi al spa e X is said to be ωα-separable if there exists a ountable subset B of X whi h is ωα-dense in X. Theorem 3.16. Let f : (X, τ) → (Y, σ) is somewhat ωα- ontinuous fun tion.If X is ωα-separable then Y is separable. Proof. Let B be ountable subset of X whi h is ωα-dense in X. Then from Theorem 3.12,f(B) is dense in Y. Sin e B is ountable f(B) is also ountable whi h is dense in Y. This implies that Y is separable. 4 Somewhat ωα-Open Fun tions In this se tion, we introdu e the on ept of somewhat ωα-open fun tions and study some of their hara terizations. De�nition 4.1. A fun tion f : (X, τ) → (Y, σ) is somewhat ωα-open provided that for open set U in X and U 6= φ there exists a ωα -open set V in Y su h that V 6= φ and V ⊆ f(U). Example 4.2. Let X = Y = {a, b, c} and τ = {X, φ, {a} , {b, c}} and σ = {X, φ, {a}}. De�ne a fun tion f : (X, τ) → (Y, σ) by f(a) = c, f(b) = a and f(c) = b. Then learly f is somewhat ωα-open. Theorem 4.3. Every somewhat open fun tion is somewhat ωα-open fun tion but onverse need not be true in general. Example 4.4. In Example 4.2, f is somewhat ωα-open fun tion but not somewhat -open fun tion. Remark 4.5. Somewhat ωα-open and somewhat semi-open fun tions are independent of ea h other as seen from the following examples. Example 4.6. In Example 4.2, f is somewhat ωα-open fun tion but not somewhat semi-open fun tion. 56 P.G.Patil, T.D. Rayanagoudar & S.S.Ben halli CUBO 15, 3 (2013) Example 4.7. Let X = Y = {a, b, c}, τ = {X, φ, {b} , {a, c}} and σ = {Y, φ, {a} , {b} , {a, b}}. Then the identity fun tion f : (X, τ) → (Y, σ) is somewhat semi-open but not somewhat ωα-open fun tion. Theorem 4.8. If f : (X, τ) → (Y, σ) is open fun tion and g : (Y, σ) → (Z.η) is somewhat ωα-open fun tion,then their omposition gof is somewhat ωα-open fun tion. We have the following hara terization. Theorem 4.9. The following statements are eqivalent for bije tive fun tion f : (X, τ) → (Y, σ) (1) f is somewhat ωα-open fun tion (2) If F is losed subset of X su h that f(F) 6= Y, then there exists a ωα- losed subset D of Y su h that D 6= Y and f(F) ⊂ D. Proof. (1) ⇒ (2):Let F be a losed subset of X su h that f(F) 6= Y. From (1), there exists a ωα -open set V 6= φ in Y su h that V ⊂ f(X − F). Put D = Y − V. Clearly D is a ωα- losed in Y and we laim that D 6= Y. If D = Y, then V = φ whi h is a ontradi tion. Sin e V ⊂ f(X − F), D = Y − V ⊂ Y − [f(X − F)] = f(F). (2) ⇒ (1):Let U be any non-empty open set in X. Put F = X − U. Then F is a losed subset of X and f(X − U) = f(F) = Y − f(U) whi h implies f(F) 6= φ. Therefore by (2) there is a ωα- losed subset D of Y su h that D 6= Y and f(F) ⊂ D. Put V = X − D, learly V is ωα-open set and V 6= φ.Further, V = X − D ⊂ Y − f(F) = Y − [Y − f(U)] = f(U). Theorem 4.10. If f : (X, τ) → (Y, σ) is somewhat ωα-open fun tion and A be any open subset of X. Thenf/A : (A, τ/A) → (Y, σ) is also somewhat ωα-open fun tion. Theorem 4.11. If f : (X, τ) → (Y, σ)be a fun tion su h that f/A and f/B are somewhat ωα-open, then fis somewhat ωα-open fun tion, where X = A ∪ B, A and B are open subsets of X. 5 Totally ωα - Continuous Fun tions In this se tion, we introdu e a new lass of fun tions alled totally ωα- ontinuous fun tions and study some of their properties. De�nition 5.1. A fun tion f : (X, τ) → (Y, σ) is said to be totally ωα- ontinuous, if the inverse image of every open subset of Y is an ωα- lopen subset of X. Example 5.2. Let X = Y = {a, b, c}, τ = {X, φ, {a}} and σ = {Y, φ, {a} , {b, c}}. De�ne a fun tion f : (X, τ) → (Y, σ) byf(a) = b, f(b) = a and f(c) = c.Then f is totally ωα- ontinuous fun tion Theorem 5.3. Every perfe tly ωα- ontinuous map is totally ωα- ontinuous but onverse need not be true in general. CUBO 15, 3 (2013) Generalization of New Continuous Fun tions in Topologi al Spa es 57 Proof. Let f : (X, τ) → (Y, σ) be a perfe tly ωα- ontinuous. Let U be an open set in Y. Then U is ωα-open in Y. Sin e f is a perfe tly ωα- ontinuous, f−1(U) is lopen in X, implies that f−1(U) is ωα- lopen in X. Example 5.4. In Example 5.2, f is totally ωα- ontinuous but not perfe tly ωα- ontinuous. Theorem 5.5. Every totally ωα- ontinuous fun tion is ωα- ontinuous but onverse need not be true in general. Example 5.6. Let X = Y = {a, b, c}, τ = {X, φ, {a}} and σ = {Y, φ, {a} , {a, c}}.Then the identity fun tion f : (X, τ) → (Y, σ) is ωα- ontinuous fun tion but not totally ωα- ontinuous fun tion. Remark 5.7. It is lear that the totally ωα- ontinuous fun tion is stronger than ωα- ontinuous and weaker than perfe tly ωα- ontinuous. Theorem 5.8. If f : (X, τ) → (Y, σ) is totally ωα- ontinuous fun tion from an ωα- onne ted spa e X in to Y, then Y is an indis rete spa e. Proof. Suppose that Y is not indis rete spa e. Let A be a proper non-empty open subset of Y. Then f−1(A) is a non-empty proper ωα- lopen subset of X whi h is ontradi tion to the fa t that X is ωα- onne ted. De�nition 5.9. A topologi al spa e X is said to be ωα2-spa e [7℄, if for every pair of distin t points x and y in X, there exists ωα-open sets M and N su h that x ∈ N , y ∈ M and M∩N = φ. Theorem 5.10. Let f : (X, τ) → (Y, σ) be totally ωα- ontinuous inje tion map. If Y is T0, then X is ωα2-spa e. Proof. Let x and y be any pair of distin t points of X. Then f(x) 6= f(y). Then there exists an open set U ontaining f(x) but notf(y). Sin e Y is T0. Then x /∈ f −1(U) and y /∈ f−1(U). Sin e f is totally ωα- ontinuous,f−1(U) is an ωα- lopen subset of X. Also x ∈ f−1(U) and y ∈ (f−1(U))c. Hen e X is ωα2-spa e. Theorem 5.11. A topologi al spa e X is ωα - onne ted if and only if every totally ωα- ontinuous fun tion from a spa e X in to any T0-spa e Y is a onstant fun tion. Theorem 5.12. Let f : (X, τ) → (Y, σ) is totally ωα- ontinuous and Y be a T1-spa e. If A is an ωα- onne ted subset of X, then f(A) is a single point. Theorem 5.13. A fun tion f : (X, τ) → (Y, σ) is totally ωα- ontinuous at a point x ∈ X if for ea h open subset V in Y ontaining f(x), there exists a ωα- lopen subset U in X ontaining x su h that f(U) ⊂ V. 58 P.G.Patil, T.D. Rayanagoudar & S.S.Ben halli CUBO 15, 3 (2013) Proof. Let V be an open subset of Y and let x ∈ f−1(V). Sin e f(x) ∈ V, there exists a ωα- lopen set Ux in X ontaining x su h that Ux ∈ f −1(V). We obtain f−1(V) = Ux∈f−1(V). Sin e arbitrary union of ωα-open sets is ωα-open, f−1(V) is ωα- lopen in X. De�nition 5.14. Let X be a topologi al spa e. Then the set of all points y in X su h that x and y annot be separated by a ωα-separation of X is said to be the quasi ωα- omponent of X. Theorem 5.15. Let f : (X, τ) → (Y, σ) is totally ωα- ontinuous map from a topologi al spa e X in to a T1-spa e Y, then f is onstant on ea h quasi ωα- omponent of X. Proof. Let x and y be two points of X that lie in the some quasi ωα- omponent of X. Assume that f(x) = α 6= β = f(y). Sin e Y is T1, α is losed in Y and so α c is an open subset in Y. Sin e f is totally ωα- ontinuous,f−1(α) and f−1(αc) are disjoint ωα- lopen subsets of X. Further x ∈ f−1(α) and y ∈ f−1(α)c, whi h is a ontradi tion in view of the fa t that y must belong to every ωα- lopen set ontaining x. Re eived: April 2013. A epted: September 2013. Referen es [1℄ S.S.Ben halli, P.G.Patil and T.D.Rayanagoudar, ωα-Closed Sets in Topologi al Spa es,The Global Jl.of Appl.Math.and Math.S ien es, V.2,1-2,(2009),53-63. 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