A Note on Reordering Ordered Topological Spaces and the Existence of Continuous, Strictly Increasing Functions

The origin of this paper is in a question that was asked of the author by Michael Wellman, a computer scientist who works in artificial intelligence at Wright Patterson Air Force Base in Dayton, Ohio. He wanted to know if, starting with R and its usual topology and product partial order, he could linearly reorder every finite subset and still obtain a continuous function from R into R that was strictly increasing with respect to the new order imposed on R. It is the purpose of this paper to explore the structural characteristics of ordered topological spaces which have this kind of behavior. In order to state more clearly the questions that we will consider, we must define some terms. By an order we mean a relation that is transitive and asymmetric (p < q ⇒ q ≮ p). Unless otherwise stated, we will denote the order of an ordered set X as <. Two distinct elements, p and q, of X are comparable if p < q or q < p, and are incomparable otherwise. An order is called a linear order if every two distinct elements of the set are comparable. A chain is a subset of an ordered set that is linearly ordered. An antichain of an ordered set X is a subset of X in which every two distinct elements are incomparable. If p and q are elements of X then [p, q] = {r ∈ X : p ≤ r ≤ q}. A subset A of X is convex if [p, q] ⊆ A for all p, q ∈ X. An ordered topological space is an ordered set with a topology. There need be no connection between the order and the topology. So R with its usual ordering (i.e. 〈a, b〉 ≤ 〈c, d〉 ⇔ a ≤ c and b ≤ d) is an ordered topological space, as is R with the usual topology and the lexicographic ordering, and R with the discrete topology and the usual ordering. We will use the word space when discussing an ordered set with a topology, and the word set when only an order is involved. Let 〈X, <X〉 and 〈Y, <Y 〉 be ordered spaces and let f : X → Y . We will say that f is strictly increasing if f(p) <Y f(q) for all p, q ∈ X with p <X q. If X and Y are linearly ordered and f is also onto, then we say that X and Y are order isomorphic. The function f is called increasing if p ≤X q implies that f(p) ≤Y f(q).