If k is a f.r. (= formally real) field which is partially ordered with positive cone, P, X P denotes the space of total orders T of k with P ⊂ T. Suppose you have a subset A ⊂ X P and an element T ∈ X P , 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 ... [Show full abstract] questions.
Question 1. Suppose A is closed. Then there is an n ∈ N and elements a 1 , …, a n ∈ k such that the basic open set H = H ( a 1 , …, a n ) 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 , …, a n )?