1. Introduction

The concept of a preopen set in a topological space was introduced by H. H. Corson and E. Michael in 1964[5]. 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[13], while the concept of a , locally dense, set was introduced by H. H. Corson and E. Michael[5].

The concept of a semi-open set in a topological space was introduced by N. Levine in 1963[12]. A subset A of a topological space (X, τ) is called semi-open[12] 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 [2]. A set A is called γ−closed if its complement is γ−open or equivalently if A ∪ S is preclosed, whenever S is preclosed. The class γ−open sets is a topology on X[1].

A real-valued function f defined on a topological space X is called A−continuous[14] 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[6,7].

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[15]. Nearly continuity or precontinuity is known also as almost continuity by T. Husain[8]. Precontinuity was studied for real-valued functions on Euclidean space by Blumberg back in 1922[3].

Results of Katˇetov[9,10] concerning binary relations and the concept of an indefinite lower cut set for a real-valued function, which is due to Brooks[4], are used in order to give a sufficient condition for the 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[11].

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: 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.

In this paper, is given a sufficient condition for the weak γ−insertion property. Also several insertion theorems are obtained as corollaries of this result.

2. The Main Result

Before giving a sufficient condition for insertability of a γ−continuous function, the necessary definitions and terminology are stated.

Let (X, τ ) be a topological space, the family of all γ−open, γ−closed, semi-open, 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 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)}.

If {Ai : i ∈ I} be a family of preopen (resp. semi-open) sets, since Ai ⊆ Int(Cl(Ai)) ⊆ Int(Cl(∪{Ai : i ∈ I})) (resp. Ai ⊆ Cl(Int(Ai)) ⊆ Cl(Int(∪{Ai : i ∈ I})), then ∪{Ai : i ∈ I}⊆ Int(Cl(∪{Ai : i ∈ I})) (resp. ∪{Ai : i ∈ I}⊆ Cl(Int(∪{Ai : i ∈ I})), i. e., ∪{Ai : i ∈ I} is a preopen (resp. semi-open) set. Therefore, both preopen and semi-open sets are preserved by arbitrary unions.

Hence, 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, pre-open.

The following first two definitions are modifications of conditions considered in[9,10].

Definition 2.2. If ρ is a binary relation in a set S then ρ¯is defined as follows: x ρ¯y if and only if y ρ v implies x ρ v 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 ρ¯ 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[4] as follows:

Definition 2.4. If f is a real-valued function defined on a space X and if {x ∈ X : f(x) 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 lower indefinite cut sets A(f, t) and A(g, t) in the domain of f and g at the level t for each rational number t such that if t1 Proof. Let g and f be real-valued functions defined on X such that g ≤ f. By hypothesis there exists a strong binary relation ρ on the power set of X and there exist lower indefinite cut sets A(f, t) and A(g, t) in the domain of f and g at the level t for each rational number t such that if t1 Define functions F and G mapping the rational numbers Qinto the power set of X by F (t)= A(f, t) and G(t)= A(g, t). If t1 and t2 are any elements of Q with t1 For any x in X, let h(x) = inf{t ∈ Q : x ∈ H(t)}.

We first verify that g ≤ h ≤ f: If x is in H(t) then x is in G(k) for any k >t; since x is in G(k)= A(g, k) implies that g(x) ≤ k, it follows that g(x) ≤ t. Hence g ≤ h. If x is not in H(t), then x is not in F (k) for any k k, it follows that f(x) ≥ t. Hence h ≤ f.

Also, for any rational numbers t1 and t2 with t1 −¹(t1,t2)= γInt(H(t2)) \ γCl(H(t1)). Hence h⁻¹(t1,t2) is a γ−open subset of X, i. e., h is a γ−continuous function on X.

The above proof used the technique of proof of Theorem 1 of[9].

3. Applications

The abbreviations pc and sc are used for precontinuous and semicontinuous, respectively.

Before stating the consequences of Theorem 2.1, we suppose that X is a topological space that γ−open sets are semi-open and preopen.

Corollary 3.1. If for each pair of disjoint preclosed (resp. semi-closed) sets F1, F2, there exist γ−open sets G1 and G2 such that F1 ⊆ G1, F2 ⊆ G2 and G1 ∩ G2 = ∅ then every precontinuous (resp. semi-continuous) function is γ−continuous.

Proof. First verify that X has the weak γ−insertion property for (pc, pc) (resp. (sc, sc)): Let g and f be real-valued functions defined on the X, such that f and g are pc (resp. sc), and g ≤ f.If a binary relation ρ is defined by AρB in case pCl(A) ⊆ pInt(B) (resp. sCl(A) ⊆ sInt(B)), then by hypothesis ρ is a strong binary relation in the power set of X. If t1 and t2 are any elements of Q with t1 A(f, t1) ⊆{x ∈ X : f(x) ≤ t1}⊆{x ∈ X : g(x) Also, if f be a real-valued precontinuous (resp. semi-continuous) function defined on the X, by setting g = f, then there exists a γ−continuous function h such that g = h = f.

Corollary 3.2. If for each pair of disjoint subsets F1,F2 of X , such that F1 is preclosed and F2 is semi-closed, there exist γ−open subsets G1 and G2 of X such that F1 ⊆ G1, F2 ⊆ G2 and G1 ∩ G2 = ∅ then X have the weak γ−insertion property for (pc, sc) and (sc, pc).

Proof. Let g and f be real-valued functions defined on the X, such that g is pc (resp. sc) and f is sc (resp. pc), with g ≤ f.If a binary relation ρ is defined by AρB in case sCl(A) ⊆ pInt(B) (resp. pCl(A) ⊆ sInt(B)), then by hypothesis ρ is a strong binary relation in the power set of X. If t1 and t2 are any elements of Q with t1 A(f, t1) ⊆{x ∈ X : f(x) ≤ t1}⊆{x ∈ X : g(x) Remark 3.1. See[1,2], for examples of topological spaces are said in corollaries 3.1 and 3.2.

ACKNOWLEDGEMENTS

This research was partially supported by Centre of Excellence for Mathematics (University of Isfahan).

MSC (2000): Primary 54C08, 54C10, 54C50; Secondary 26A15, 54C30.