Companions of directed sets and the Ordering Lemma

Abstract

Given a partially ordered set (D,≤), a companion (C⪯) of (D,≤) is a well ordered set where C is a cofinal subsets of (D,≤) such that for every c1,c2∈C if c1≤c2 then c1⪯c2. The Ordering Lemma says that every partially ordered set has a companion. Given a directed set (D,≤) and a net f:D→X, the restriction f↾C of the net to the companion (C,⪯) of (D,≤) is a transfinite sequence. We show how the convergence and clustering of f↾C is related to the convergence and clustering of f.