Strong Insertion of A Baire-one Function

Majid Mirmiran

doi:10.13189/ms.2013.010203

Abstract

Necessary and sucient conditions in terms of lower cut sets are given for the strong insertion of a Baire-one function between two comparable real- valued functions on the topological spaces that sets 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 Baire-one function h such that g h f and such that if g(x) < f(x) for any x in X, then g(x) < h(x) < f(x). In this paper, for a topological space that sets are G sets, is given a sucient condition for the weak B1insertion property. Also for a space with the weak B1insertion property, we give necessary and sucient conditions for the space to have the strong B1insertion property. Several insertion theorems are obtained as corollaries of these results.

Highlights

A generalized class of closed sets was considered by Maki in 1986 [7]
He investigated the sets that can be represented as union of closed sets and called them V −sets
Results of Katetov [3], [4] 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 a Baire-one function between two comparable real-valued functions on the topological spaces that Λ−sets are Gδ−sets

Summary

Introduction

A generalized class of closed sets was considered by Maki in 1986 [7]. He investigated the sets that can be represented as union of closed sets and called them V −sets. A real-valued function f defined on a topological space X is called Baire-one if the preimage of every open subset of R is a Fσ−set in X. If P1 and P2 are B1−properties, the following terminology is used: (i) A space X has the weak B1−insertion property for (P1, P2) iff for any functions g and f on X such that

Results

Conclusion