Local Stability and Saturation in Spaces of Orderings

Abstract

If k is a f.r. (= formally real) field which is partially ordered with positive cone, P, XP denotes the space of total orders T of k with P ⊂ T. Suppose you have a subset A ⊂ XP and an element T ∈ XP , T ∉ A. Then the main question investigated in this paper is the following: How can T be separated from A by using elements of k? To be more specific, this is split up into two different questions. Question 1. Suppose A is closed. Then there is an n ∈ N and elements a 1, …, an ∈ k such that the basic open set H = H(a 1, …, an ) is a neighborhood of T and has empty intersection with A. Now, if T is given, what is the least n ∊ N (if it exists) such that T has a neighborhood basis consisting of basic open sets of the form H(a 1, …, an )?