Profinite separation systems
Reinhard Diestel, Jay Lilian Kneip
TL;DR
This paper develops the theory of profinite separation systems as inverse limits of finite systems, equipping them with a topology and a framework to transfer finite tangle-duality results to the infinite setting. It establishes foundational relations between a profinite system and its finite projections, introduces notions like essential closure and essential-over, and proves a tree-set compactness theorem that enables lifting nested structures from finite levels to the profinite level. It also analyzes how orientations of profinite nested systems behave, culminating in a criterion for normality in profinite tree sets. These results set the stage for tangle duality theorems for infinite graphs and matroids, to be developed in subsequent work.
Abstract
Separation systems are posets with additional structure that form an abstract setting in which tangle-like clusters in graphs, matroids and other combinatorial structures can be expressed and studied. This paper offers some basic theory about infinite separation systems and how they relate to the finite separation systems they induce. They can be used to prove tangle-type duality theorems for infinite graphs and matroids, which will be done in future work that will build on this paper.