Type: Article

Insertion theorems for maps to linearity ordered topological spaces

García-Ferreira, S; Ortiz-Castillo, YF; Yamauchi, T.;

Abstract:

Let X be a topological space and (Y, <=) a linearly ordered topological space. Following the Katetov-Tong Insertion Theorem, a pair (X, Y) is said to have the insertion property if for every upper semicontinuous map g : X -> Y and every lower semicontinuous map h : X -> Y with g(x) <= h(x) for all x is an element of X, there exists a continuous map f : X -> Y such that g(x) <= f (x) <= h(x) for all x is an element of X. We show that if (X, Y) has the insertion property for every normal space X, then Y is order isomorphic to some interval in the real line. We also prove that if X is a paracompact space or a cardinal and L is the double edged long line, then (X, L) has the insertion property