Strong insertion of a $\gamma-$ continuous function

Bol. Soc. Paran. Mat. c SPM –ISSN-2175-1188 on line SPM: www.spm.uem.br/bspm (3s.) v. 34 2 (2016): 43–52. ISSN-00378712 in press doi:10.5269/bspm.v34i2.19642 Strong Insertion of a γ−continuous Function Majid Mirmiran and Binesh Naderi ∗ abstract: Necessary and sufficient conditions in terms of lower cut sets are given for the strong insertion of a γ−continuous function between two comparable real-valued functions. Key Words: Strong insertion, Strong binary relation, Preopen set, Semi-open set, γ−open set. Contents 1 Introduction 43 2 The Main Result 44 3 Applications 46 1. Introduction The concept of a preopen set in a topological space was introduced by H. H. Corson and E. Michael in 1964 [4]. A subset A of a topological space (X, τ ) is called preopen or locally dense or nearly open if A ⊆ Int(Cl(A)). A set A is called preclosed if its complement is preopen or equivalently if Cl(Int(A)) ⊆ A. The term ,preopen, was used for the first time by A. S. Mashhour, M. E. Abd El-Monsef and S. N. El-Deeb [12], while the concept of a , locally dense, set was introduced by H. H. Corson and E. Michael [4]. The concept of a semi-open set in a topological space was introduced by N. Levine in 1963 [11]. A subset A of a topological space (X, τ ) is called semi-open [11] if A ⊆ Cl(Int(A)). A set A is called semi-closed if its complement is semi-open or equivalently if Int(Cl(A)) ⊆ A. Recall that a subset A of a topological space (X, τ ) is called γ−open if A ∩ S is preopen, whenever S is preopen [1]. A set A is called γ−closed if its complement is γ−open or equivalently if A ∪ S is preclosed, whenever S is preclosed. we have that if a set is γ−open then it is semi-open and preopen. Recall that a real-valued function f defined on a topological space X is called A−continuous [15] if the preimage of every open subset of R belongs to A, where A is a collection of subset of X. Most of the definitions of function used throughout this paper are consequences of the definition of A−continuity. However, for unknown concepts the reader may refer to [5,6]. ∗ This work was supported by University of Isfahan and Centre of Excellence for Mathematics (University of Isfahan). 2000 Mathematics Subject Classification: Primary 54C08; Secondary 26A15, 54C10, 54C30, 54C50. 43 Typeset by BSP style. M c Soc. Paran. de Mat. 44 Majid Mirmiran and Binesh Naderi Hence, a real-valued function f defined on a topological space X is called precontinuous (resp. semi-continuous or γ−continuous) if the preimage of every open subset of R is preopen (resp. semi-open or γ−open) subset of X. Precontinuity was called by V. Ptak nearly continuity [16].Nearly continuity or precontinuity is known also as almost continuity by T. Husain [7].Precontinuity was studied for real-valued functions on Euclidean space by Blumberg back in 1922 [2]. Results of Katětov [8,9] concerning binary relations and the concept of an indefinite lower cut set for a real-valued function, which is due to Brooks [3], are used in order to give necessary and sufficient conditions for the strong insertion of a γ−continuous function between two comparable real-valued functions. If g and f are real-valued functions defined on a space X, we write g ≤ f in case g(x) ≤ f (x) for all x in X. The following definitions are modifications of conditions considered in [10]. A property P defined relative to a real-valued function on a topological space is a γ−property provided that any constant function has property P and provided that the sum of a function with property P and any γ−continuous function also has property P . If P1 and P2 are γ−property, the following terminology is used:(i) A space X has the weak γ−insertion property for (P1 , P2 ) if and only if for any functions g and f on X such that g ≤ f, g has property P1 and f has property P2 , then there exists a γ−continuous function h such that g ≤ h ≤ f .(ii) A space X has the strong γ−insertion property for (P1 , P2 ) if and only if for any functions g and f on X such that g ≤ f, g has property P1 and f has property P2 , then there exists a γ−continuous function h such that g ≤ h ≤ f and if g(x) < f (x) for any x in X, then g(x) < h(x) < f (x). In this paper, we give a sufficient condition for the weak γ−insertion property. Also for a space with the weak γ−insertion property, we give necessary and sufficient conditions for the space to have the strong γ−insertion property. Several insertion theorems are obtained as corollaries of these results. In addition, the insertion of a γ−continuous function has also considered by the author in [14] 2. The Main Result Before giving a sufficient condition for insertability of a γ−continuous function, the necessary definitions and terminology are stated. The abbreviations pc , sc and γc are used for precontinuous , semicontinuous and γ−continuous, respectively. Let (X, τ ) be a topological space, the family of all γ−open, γ−closed, semiopen, semi-closed, preopen and preclosed will be denoted by γO(X, τ ), γC(X, τ ), sO(X, τ ), sC(X, τ ), pO(X, τ ) and pC(X, τ ), respectively. Definition 2.1. Let A be a subset of a topological space (X, τ ). Respectively, we define the γ−closure, γ−interior, s-closure, s-interior, p-closure and p-interior Strong Insertion of a γ−continuous Function 45 of a set A, denoted by γCl(A), γInt(A), sCl(A), sInt(A), pCl(A) and pInt(A) as follows: γCl(A) = ∩{F : F ⊇ A, F ∈ γC(X, τ )}, γInt(A) = ∪{O : O ⊆ A, O ∈ γO(X, τ )}, sCl(A) = ∩{F : F ⊇ A, F ∈ sC(X, τ )}, sInt(A) = ∪{O : O ⊆ A, O ∈ sO(X, τ )}, pCl(A) = ∩{F : F ⊇ A, F ∈ pC(X, τ )} and pInt(A) = ∪{O : O ⊆ A, O ∈ pO(X, τ )}. Respectively, we have γCl(A), sCl(A), pCl(A) are γ−closed, semi-closed, preclosed and γInt(A), sInt(A), pInt(A) are γ−open, semi-open, preopen. The following first two definitions are modifications of conditions considered in [8,9]. Definition 2.2. If ρ is a binary relation in a set S then ρ̄ is defined as follows: x ρ̄ y if and only if y ρ ν implies x ρ ν and u ρ x implies u ρ y for any u and v in S. Definition 2.3. A binary relation ρ in the power set P (X) of a topological space X is called a strong binary relation in P (X) in case ρ satisfies each of the following conditions: 1) If Ai ρ Bj for any i ∈ {1, . . . , m} and for any j ∈ {1, . . . , n}, then there exists a set C in P (X) such that Ai ρ C and C ρ Bj for any i ∈ {1, . . . , m} and any j ∈ {1, . . . , n}. 2) If A ⊆ B, then A ρ̄ B. 3) If A ρ B, then γCl(A) ⊆ B and A ⊆ γInt(B). The concept of a lower indefinite cut set for a real-valued function was defined by Brooks [3] as follows: Definition 2.4. If f is a real-valued function defined on a space X and if {x ∈ X : f (x) < ℓ} ⊆ A(f, ℓ) ⊆ {x ∈ X : f (x) ≤ ℓ} for a real number ℓ, then A(f, ℓ) is called a lower indefinite cut set in the domain of f at the level ℓ. We now give the following main result: Theorem 2.1. Let g and f be real-valued functions on a topological space X with g ≤ f . If there exists a strong binary relation ρ on the power set of X and if there exist l (...truncated)