Strong Insertion of an Α - Continuous Function 1

The concept of a preopen set in a topological space was introduced by H. H. Corson and E. Michael in 1964 [3]. 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 [11], while the concept of a, locally dense, set was introduced by H. H. Corson and E. Michael [3]. The concept of a semi-open set in a topological space was introduced by N. Levine in 1963 [10]. A subset A of a topological space (X, τ) is called semi-open [10] 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 α-open if A is the difference of an open and a nowhere dense subset of X. A set A is called α-closed if its complement is α-open or equivalently if A is union of a closed and a nowhere dense set. We have a set is α-open if and only if it is semi-open and preopen. Recall that a real-valued function f defined on a topological space X is called A-continuous [13] 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 used throughout this paper are consequences of the definition of A-continuity. However, for unknown concepts the reader may refer to [4, 5]. Hence, a real-valued function f defined on a topological space X is called precontinuous (resp. semi-continuous or α-continuous) if the preimage of every open subset of R is preopen (resp. semi-open or α-open) subset of X. Precontinuity was called by V. Pták nearly continuity [14]. Nearly continuity or precontinuity is known also as almost continuity by T. Husain [6]. Precontinuity was studied for real-valued functions on Euclidean space by Blumberg back in 1922 [1]. Results of Katětov [7, 8] concerning binary relations and the concept of an indefinite lower cut set for a real-valued function, which is due to Brooks [2], are used in order to give necessary and sufficient conditions for the strong insertion of an α-continuous function between two comparable real-valued functions.