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Advances in Mathematics and Computer Science and their Applications CHARACTERIZATIONS OF FEEBLY TOTALLY OPEN FUNCTIONS Raja Mohammad Latif Department of Mathematics and Natural Sciences Prince Mohammad Bin Fahd University P.O. Box 1664, Al – Khobar 31952, Kingdom of Saudi Arabia E-Mails: [email protected] & [email protected] Mohammad Rafiq Raja Department of Mathematics, University of Sargodha Mandi Bahauddin Campus, Pakistan E-Mails: [email protected] & [email protected] Muhammad Razaq Department of Mathematics, Horizon Degree and Commerce College, Bhaun Road, Chakwal, Pakistan E-Mail: [email protected] Abstract: In this paper, feebly closed and feebly open sets are used to define and investigate a new class of functions called, feebly totally open functions and also study some of their basic properties. Relationships between this new class and other classes of existing known functions are established. 2010 Mathematical Subject Classification: 54C10 Keywords: Topological Space, feebly open set, feebly closed set, clopen set, feebly totally open (closed) function, feebly totally continuous function. ISBN: 978-1-61804-360-3 217 Advances in Mathematics and Computer Science and their Applications 1. Introduction closure of A, denoted by FCl(A), is the intersection of all feebly closed sets containing A. If (X, τ) is a space, then FO(X, τ) = {F⊆X : F is feebly open set in X} is a topology on X, τ⊆FO(X, FO(X, τ)) = FO(X, τ), and for each U⊆X, sClτ (U ) = sClFO( X , τ ) (U ) [13]. If (X, τ) is Throughout this paper by a space (X, τ) we mean a topological space. If A is any subset of a space X, then Cl(A) and Int(A) denote the closure and the interior of A respectively. Njastad [40] introduced the concept of an α-set in (X, τ). A subset A of (X, τ) is called an α-set if A⊆Int[Cl(Int(A))]. The notion of semi-open set and pre-open set were introduced by Levine [28] and Mashhour et al [32] respectively. A subset A of (X, τ) is called a semi-open set (respectively pre-open set) if A⊆Cl[(Int(A))] (respectively A⊆Int[(Cl(A))]. The complement of an α-set, a semi-open set and a pre-open set are called α-closed, semi-closed and pre-closed respectively. We denote the family of all α-sets in (X, τ) by α(X, τ). Njastad [40] proved that α(X, τ) is a topology on X. If A is a subset of (X, τ), then the intersection of all semi-closed sets containing A is called the seni-closure of A, and is denoted by sCl(A). The largest semiopen set contained in A is denoted by sInt(A). In 1963 Levine [28] showed that since the semiopen set of a topological space need not be closed under finite intersection, then the collection of all semi-open sets need not be a topology on the base set. This result raised questions about the collection of feebly open sets for a given topological space, which led to the following discoveries. In [10] & [35] the semi-closure of sets was used to define and investigate feebly open sets, feebly closed sets, and the feebly closure of sets. Maheshwari and Tapi [36] defined A to be a feebly open set in (X, τ) if there is an open set U such that U⊆A⊆sCl(U). Equivalently [26] A is an α-set if and only if A is feebly open set if and only if A⊆Int[Cl(Int(A))] and A is an α-closed set if and only if A is feebly closed if and only if Cl[Int(Cl(A))]⊆A if and only if sInt[Cl(A)]⊆A. Evidently every feebly open set is semi-open as well as pre-open set. The complement of feebly open set is called a feebly closed set. The feebly ISBN: 978-1-61804-360-3 a topological space, then FO(X, τ) is called the feebly induced topology. In [13] & [15] several properties of topological spaces were investigated and characterized using the feebly induced topology of a given space. Feebly continuous is one of the important properties for studying topological spaces. Several types of feebly continuous functions occur in the literature, many authors used the concepts of feebly open and feebly closed sets to study other concepts and gave several results about that. In 1980, Jain [25] introduced totally continuous functions. T.M. Nour [43] introduced the concept of totally semicontinuous functions as a generalization of totally continuous functions and several properties of totally semi-continuous functions were obtained. In [6] Benchalli and Neeli introduced semi-totally continuous and semitotally open functions and investigated some basic properties and characteristics of these functions. In this paper, feebly closed and feebly open sets are used to define and investigate a new class of functions called, feebly totally open functions. We study some of their basic properties. We will also establish some relationships between this new class of functions and other classes of existing known functions. 2. PRELIMINARIES Definition 2.1. A topological space (X, τ) is said to be: 1. semi-T₀ [33] if for each pair of distinct points in X, there exists a semi-open set containing 218 Advances in Mathematics and Computer Science and their Applications totally continuous [25] if the inverse image of every open subset of Y is a clopen subset of X. 3. strongly continuous [44] if the inverse image of every subset of Y is a clopen subset of X. 4. totally semi-continuous [43] if the inverse image of every open subset of Y is semi-clopen in X. 5. strongly semi-continuous [43] if the inverse image of every subset of Y is semiclopen in X. 6. pre-semi-open [43] if the image of every semi-open set in X is semi-open in Y. 7. semi-totally continuous [6] if the inverse image of every semi-open set in Y is clopen in X. 8. [13] feebly-continuous ( f − continuous) function if the inverse image of every closed (open) set in Y is feebly closed ( f − open) set one point but not the other. 2. sT1/2 [6] if every semi-closed set of X is closed in X. 3. semi-T₁ [33] (resp. clopen T₁ [22]) if for each pair of distinct points x and y of X, there exist semiopen (clopen) sets U and V containing x and y respectively such that y∉U and x∉V. 4. semi-T₂ [33] & [26] (resp. ultra Hausdorff or UT₂ [45]) if every two distinct points of X can be separated by disjoint semi-open (resp. clopen) sets. 5. s-normal [34] (resp. ultra-normal [33 & 45]) if each pair of non-empty disjoint closed sets can be separated by disjoint semi-open (resp. clopen) sets. 6. s-regular [26] (resp. ultra regular [33]) if for each closed set F of X and each x ∉ F, there exist disjoint semi-open (resp. clopen) sets U and V such that F⊆U and x∈V. 7. semi-normal [14] (resp. clopen normal [22]) if for each pair of disjoint semi-closed (resp. clopen) sets U and V of X, there exist two disjoint semi-open (resp. open) sets G and H such that U⊆G and V⊆H. 8. semi-regular [12] (resp. clopen regular [22]) if for each semiclosed (resp. clopen) set F of X and each x∉F, there exist disjoint semi-open (resp. open) sets U and V such that F⊆U and x∈V. 9. locally indiscrete [41] if every open set of X is closed in X. 10. s-connected [36] if X is not the union of two nonempty disjoint semi-open subsets of X. 11. feebly T₁ [26] if for each pair of distinct points x, y in X, there exist feebly open sets U and V containing x and y respectively such that y∉U and x∉V. 12. [33] clopen-T₁ if for each pair of distinct points x and y of X, there exist clopen sets U and V containing x and y respectively such that y∉U and x∉V. 13. [33] ultra – normal if each pair of nonempty disjoint closed sets can be separated by clopen sets. in X. 9. [3] f * − continuous if the inverse image of every feebly open (closed) set in Y is open (closed) set in X. 10. [3] f ** − continuous if the inverse image of every feebly open (closed) set in Y is feebly open (closed) set in X. 11. [24] feebly-closed ( f − closed) [ feebly-open ( f − open)] function if the image of every closed (open) set in X is feebly closed (feebly open) set in Y. 12. semi-open [7] if f (U ) is semiopen in Y for each open set U in X. 13. semiclosed [8] if f (U ) is semi-closed in Y for each closed set F in X. Definition 2.3. Let f : (X, τ) → (Y, σ) be a function. Then the graph function of f defined by g ( x ) = ( x, f ( x ) ) for each x∈X. Definition 2.4. [43] Let (X, τ) be a topological space and x∈X. Then the set of all points y in X such that x and y cannot be separated by semiseparation of X is said to be the quasi semicomponent of x. A quasi semi-component of a point x in a space X means the intersection of all semi-clopen sets containing x. Definition 2.2. Let f : (X, τ) →(Y, σ) be a function. Then f is said to be: 1. semi-continuous [28] if the inverse image of each open subset of Y is semi-open in X. 2. ISBN: 978-1-61804-360-3 is 219 Advances in Mathematics and Computer Science and their Applications 3. FEEBLY TOTALLY OPEN FUNCTIONS This implies, f ( F )= Y − V , which is clopen in Definition 3.1. A function f : (X, τ) →(Y, σ) is said to be feebly totally open (closed) if the image of every feebly open (closed) set in X is clopen in Y. Y. Thus, the image of a feebly open set in X is clopen in Y. Therefore f is a feebly totally open function. Theorem 3.5. For any bijective function f : (X, τ) →(Y, σ), the following statements are equivalent: Theorem 3.2. Let f : (X, τ) →(Y, σ) be a bijective function. Then f is feebly totally open if and only if f is feebly totally closed function. (i) Inverse of f is feebly totally continuous. (ii) f is feebly totally open. Proof. Suppose that f is a feebly totally open (closed) funtion. Let F be a feebly closed (feebly open) set in X. Then X − F is feebly open (feebly closed) in X. Since f is feebly Proof. (i) ⇒ (ii): Let U be a feebly open set of X. By assumption ( f −1 (U ) ) = f (U ) is clopen −1 in Y . So f is feebly totally open. totally open (closed), f ( X − F ) =Y − f ( F ) is clopen in Y. This implies f ( F ) is clopen in Y. (ii) ⇒ (i): Let V be a feebly open set in X. Then Theorem 3.3. A surjective function f : (X, τ) →(Y, σ) is feebly totally open if and only if for each subset B of Y and for each feebly closed f (V ) is ( f (V ) ) −1 −1 clopen in Y. That is = f (V ) is clopen in Y. Therefore f −1 is feebly totally continuous. set U containing f −1 ( B ) , there exists a clopen Theorem 3.6. If f : (X, τ) → (Y, σ) is feebly totally continuous surjection from a feebly normal space X to a space Y, then Y is ultranormal set V of Y such that B⊆V and f −1 (V ) ⊆ U . Theorem 3.4. Suppose f : (X, τ) →(Y, σ) is a surjective feebly totally open function and B⊆Y. Let U be feebly closed set of X such that is a Proof. Let A and B be disjoint closed subsets of Y. Since f : X → Y is feebly totally clopen subset of Y containing B such that continuous function, f −1 ( A ) and f −1 ( B ) are Y − f ( X −U ) f −1 ( B ) ⊆U. Then V = f −1 (V ) ⊆ U . disjoint clopen hence disjoint closed sets in X. Since X is feebly normal, there exist disjoint feebly open sets U and V such that f −1 ( A ) ⊆U On the other hand, suppose F is a feebly closed set of X. Then f −1 (Y − f ( F ) ) ⊆ ( X − F ) and and f −1 ( B ) ⊆V. Let G = f −1 ( A ) and H = X − F is feebly open. By hypothesis, there exists a clopen set V of Y such that Y − f ( F ) ⊆ V , f −1 (V ) ⊆ ( X − F ) . which f −1 ( B ) . Clearly A⊆G, B⊆H and f −1 ( G ) ⊆U, f −1 ( H ) ⊆V. Then we have, f −1 ( G ) ∩ f −1 ( H ) implies ⊆U∩V = ϕ, which implies f −1 ( G ∩ H ) ⊆ ϕ, Therefore F ⊆ X − f (V ) . which implies G∩H = ϕ. Thus every pair of nonempty disjoint closed sets in Y can be Hence (Y − V ) ⊆ f ( F ) ⊆ f X − f −1 (V ) ⊆ (Y − V ) . ISBN: 978-1-61804-360-3 220 Advances in Mathematics and Computer Science and their Applications separated by disjoint clopen sets. Therefore Y is ultra-normal. Proof. Let V be a feebly open set in Z. Since g is f ** − continuous, g −1 (V ) is feebly open set in Y. Now since f is feebly totally continuous, Theorem 3.7. Let f : (X, τ) → (Y, σ) and g : (Y, σ) → (Z, δ) be feebly totally open functions, then g f : X → Z is feebly totally open. f −1 g −1 (V ) = ( gο f ) Hence gο f continuous. Proof. Let V be any feebly open set in X. Since f is feebly totally open, f (V) is clopen in Y. Z. Hence g f is feebly totally open. Theorem 3.8. Let f : (X, τ) → (Y, σ) and g : (Y, σ) → (Z, δ) be two functions such that g f : X → Z is feebly totally open function. : X → Z is feebly totally Conversely, let g f : X → Z be feebly totally continuous. Let W be feebly open set in Z. Since g f : X → Z is feebly totally continuous, ( g f ) (W ) = −1 f −1 g −1 (W ) is clopen in X. Since f is feebly totally open bijection, Proof. (i) Let V be a feebly open set in Y. Then f −1 (V ) is feebly open set in X because ( ) −1 = f ( g f ) (W ) f= f −1 g −1 (W ) g −1 (W ) is feebly open set in Y. Thus the inverse image of each feebly open set in Z is feebly open in Y. ( g f ) is feebly ( g f ) ( f −1 (V ) ) = g (V ) is clopen f is f − continuous. Since Hence g is f ** − continuous. in Z. This shows that g is feebly totally open. (ii) Since g is injective, we have, f ( A ) = g −1 ( ( g f )( A ) ) is true for every subset ACKNOWLEDGEMENT The first author is highly and gratefully indebted to the Prince Mohammad Bin Fahd University, Al – Khobar, Saudi Arabia, for providing all necessary research facilities during the preparation of this research paper. A of X. Let U be any feebly open set in X. Therefore ( g f )(U ) is clopen and hence open in Z. Since g is totally continuous, g −1 ( ( g f )(U ) ) = f (U ) is clopen in Y. This shows that f is feebly totally open. Theorem 3.9. If f : (X, τ) → (Y, σ) is feebly totally continuous and g : (Y, τ) → (Z, η) is f ** − continuous, then gο f : X → Z is feebly totally continuous. ISBN: 978-1-61804-360-3 is clopen in X. Proof. Let g : Y → Z be f ** − continuous. Then the proof follows from Theorem 3.9. Then (i) If f is f ** − continuous and surjective, then g is feebly totally open. (ii) If g is totally continuous and injective, then f is feebly totally open. totally open (V ) Theorem 3.10. Let f : (X, τ) → (Y, σ) be feebly totally open bijection and g : (Y, σ) → (Z, δ) be any function. Then g f : X → Z is feebly totally continuous if and only if g is f ** − continuous. Since each clopen set is feebly open set. So f (V) is feebly open set in Y. Since g is feebly totally open, g ( f (V ) ) = ( g f )(V ) is clopen in ** −1 221 Advances in Mathematics and Computer Science and their Applications 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 13. Dorsett, C. (1985). Feeble Separation Axioms, the Feebly Induced Topology, and Separation Axioms and the Feeble Induced Topology. Karadeniz University Mathematical Journal, Vol. 8, 43 – 54. 14. Dorsett, C. (1985). Semi-normal spaces, Kyungpook Math. J. Vol. 25, 173 – 180. 15. Dorsett, C. (1985). New Characterizations of Topological Properties Using Regular Open Sets and r-topological Properties. Bulletin of The Faculty of Science, Assiut University. A. Physics and Mathematics, 14(1), 75 – 88. 16. Devi, R., Balachandran, K. and Maki, H. (1995). Semi-generalized homeomorphisms and generalized semi-homeomorphisms in topological spaces, India J. Pure. Appl. Math., Vol. 26, 271 – 284. 17. Ibraheem, D. (2007). On Generalized Feebly closed sets, First Scientific Conference of College of Science, AlMuthana University. 18. Ibrahim, D. (Sep. 2008). On generalized feebly closed sets, Uruk Refereed Journal for Scientific Studies Issued by AlMuthana University, No. 1, 131 – 139. 19. Dontchev, J. (1995). On generalizing semipreopen sets, Mem. Fac. Sci. Kochi Univ. (Math.), 16, 35 – 48. 20. Dorsett, C. (1982). Semi-regular spaces, Soochow J. Math., Vol. 8, 45 – 53. 21. Ellis, R.L. (1967). A non-Archimedean analogue of the Tietze-Urysohn extension theorem, Nederl. Akad. Wetensch. Proc. Ser. A, Vol. 70, 332 – 333. 22. Ekici, E. and Caldas, M. (2004). Slightlycontinuous functions, Bol. Soc. Paran. Mat., 22(2), 63 – 74. 23. Gnanambal, Y. (1997). On generalized pre-regular closed sets in topological spaces, Indian J. Pure. Appl. Math., 28(3), 351 – 360. 24. Greenwood, S. and Reilly, I. L. (1986). September). On feebly closed mappings, REFERENCES Abd El-Monsef, M.E., El-Deeb, S.N. and Mahmoud, R.A. (1983). β- open sets, βcontinuous mappings, Bull. Fac. Sc. Assuit Univ., Vol. 12, 77 – 90. Abd El-Monsef, M.E., Kozae, A.M. and Abu-Gdairi, R.A. (2010). New approaches for generalized continuous functions, Int. Journal of Math. Analysis, 4(27), 1329 – 1339. Al-Azawi, S.N., Al-Obaidi, J.M. and Saied, A.S. (2008). On feebly continuous functions and feebly compact spaces, Diala, Jour, Vol. 29. Andrijevic, D. (1986). Semi-preopen sets, Mat.Vesnik, 38(1), 24 – 32. Arya, S.P. and Nour, T.M. (1990). Characterizations of s- normal spaces, Indian J. Pure Appl. Math., 21(8), 717 – 719. Benchalli, S.S. and Neeli, U.I. (2011). Semi-Totall Continuous Functions in Topological Spaces, International Mathematical Forum, 6(10), 479 – 492. Biswas, N. (1969). Some mappings in topological spaces, Bull. Cal. Math. Soc., Vol. 61, 127 – 135. Biswas, N. (1970). Characterization of semi-continuous mappings, Atti. Accad. Naz. Lience. Rend.Cl. Sci. Fis. Mat. Nat., 48(8), 399 – 402. Balachandran, K., Sundaram, P. and Maki, H. (1991). On generalized continuous functions in topological spaces, Mem. Fac. Sci. Kochi Univ. Ser. A Math., Vol. 12, 5 – 13. Crossley, S.G. and Hildebrand, S.K. (1971). Semi-closure, Texas J. Sci., Vol. 22, 99 – 112. 11. Crossley, S.G. and Hildebrand, S.K. (1972). Semi-Topological properties, Fund. Math., Vol. 74, 233 – 254. 12. Dorsett, C. (1982). Semi-regular spaces, Soochow J. Math., Vol. 8, 45 – 53. ISBN: 978-1-61804-360-3 222 Advances in Mathematics and Computer Science and their Applications Indian J. Pure Appl. Math., 17(9), 1101 – 1105. 25. Jain, R.C. (1980). The role of regularly open sets in general topology, Ph.D. Thesis, Meerut University, Institute of advanced studies, Meerut-India. 26. Jankovic, D.S. and Reilly, I.L, (Sep. 1985). On some separation properties, Indian J. Pure Appl. Math., 16(9), 957 – 964. 27. Kelley, J. (1955). General topology, Van Nostrand Company. 28. Levine, N. (1963). Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly, Vol. 70, 36 – 41. Fac. Sci. Kochi Univ. (Math.), Vol. 17, 33 – 42. 38. Mashhour, A.S., Abd El-Monsef, M.E. and El-Deeb, S. N. (1982). On pre continuous and weak precontinuous mappings, Proc. Math. Phys. Soc. Egypt, Vol. 53, 47 – 53. 39. Mashhour, A.S., Hasanein, I.A. and El – Deeb, S.N. (1983). α- continuous and αopen functions, Acta Math Hung., 41(3 – 4), 213 – 218. 40. Njastad, O. (1965). On some classes of nearly open sets, Pacific J. Math. Vol. 15, 961 – 970. 41. Nieminen, T. (1977). On ultrapseudo compact and related spaces, Ann. Acad. Sci. Fenn. Ser. A I Math., Vol. 3, 185 – 205. 42. Noiri, T., Maki, H. and Umehara, J. (1998). Generalized preclosed function, Mem. Fac. Sci. Kochi Univ. (Math.), Vol. 19, 13 – 20. 43. Nour, T.M. (1995). Totally semicontinuous functions, Indian J. Pure Appl. Math., 26(7), 675 – 678. 44. Stone, M. (1937). Applications of the theory of boolean rings to general topology, Trans. Amer. Math. Soc., vol. 41, 374 – 481. 45. Staum, R. (1974). The algebra of bounded continuous functions into a nonarchimedean field, Pacific J. Math., Vol. 50, 169 – 185. 46. Sundaram, P., Maki, H. and Balachandran, K. (1991). Semi-generalized continuous maps and semi- T1/2 spaces, Bull. Fukuoka 29. Levine, N. (1970). Generalized closed sets in topology, Rend. Circ. math. palermo, 19(2), 89 – 96. 30. Levine, N. (1964). Simple extension of topologies, Amer. Math. Japan. Monthly, Vol. 71, 22 – 105. 31. Maki, H., Devi, R. and Balachandran, K. (1994). Associated topologies of generalized α- closed sets and αgeneralized closed sets, Mem. Fac, Sci. Kochi Univ. (Math.), Vol. 15, 51 – 63. 32. Maheshwari, S.N. and Jain, P.C. (1982). Some new mappings, Mathematica, 24(47) (1 – 2), 53 – 55. 33. Maheshwari, S.N. and Prasad, R. (1975). Some new separation axioms, Ann. Soc. Sci. Bruxelles., 89(3), 395 – 407. 34. Maheshwari, S.N. and Prasad, R. (1978). On s-normal spaces, Bull. Math. Soc. Sci. Math. R. S. Roumanie, 22(68), 27 – 29. 35. Maheshwari, S.N. and Tapi, U.D. (1978 – 1979). Note on Some Applications on Feebly Open Sets, Mahya Bharati, J. Univ. Saugar. 36. Maheshwari, S.N. and Tapi, U.D. (1979). Connectedness of a stronger type in topological spaces, Nanta. Math., Vol. 12, 102 – 109. 37. Maki, H., Umehara J. and Noiri, T. (1996). Every topological space is pre- T1/2 , Mem. ISBN: 978-1-61804-360-3 Univ. Ed. Part III, Vol. 40, 33 – 40. 47. Wilansky, A. (2008). Topology for Analysis. Dover Publications. 48. Willard, S. (1970). General Topology. Dover Publications, Inc. Mineola, New York. 223