Jacobs Journal of Computer Science

Strong Insertion of a Contra Alpha Continuous Function

*Majid Mirmiran
Department Of Mathematical Sciences, University Of Isfahan, Iran, Islamic Republic Of

*Corresponding Author:
Majid Mirmiran
Department Of Mathematical Sciences, University Of Isfahan, Iran, Islamic Republic Of
Email:mirmir@sci.ui.ac.ir

Published on: 2019-05-13

Abstract

Necessary and sufficient conditions in terms of lower cut sets are given for the insertion of a contra-α−continuous function between two comparable real-valued functions.

Keywords

Insertion; Strong binary relation; Semi-open set; Preopen set; ??open set; Lower cut set

Introduction

The concept of a preopen set in a topological space was introduced by H.H. Corson and E. Michael in 1964. 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 pre-closed 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, while the concept of a, locally dense, set was introduced by H.H. Corson and E. Michael. The concept of a semi-open set in a topological space was introduced by N. Levine in 1963. A subset A of a topological space (X,τ) is called semiopen 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.