Insertion theorems for maps to linearly ordered topological spaces

Abstract

Let X be a topological space and (. Y, ≤) a linearly ordered topological space. Following the Katětov-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∈. X, there exists a continuous map f:. X→. Y such that g(. x). ≤. f(. x). ≤. h(. x) for all x∈. 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.