Representations of Infinite Tree Sets

Abstract

Tree sets are abstract structures that can be used to model various tree-shaped objects in combinatorics. Finite tree sets can be represented by finite graph-theoretical trees. We extend this representation theory to infinite tree sets. First we characterise those tree sets that can be represented by tree sets arising from infinite trees; these are precisely those tree sets without a chain of order type ω + 1. Then we introduce and study a topological generalisation of infinite trees which can have limit edges, and show that every infinite tree set can be represented by the tree set admitted by a suitable such tree-like space.

Trees of tangles in infinite separation systems

“…Such a method already exists in finite graphs [2]. The ingredients of that proof together with the results of [17] are all that is needed to show an analogous result for infinite graphs, which we shall present here.…”

“…Let T = (V, E) be the tree from Theorem 2•6. Note that by [17,Theorem 3•9(iii)] any isomorphism between the edge tree sets of two distinct trees induces an isomorphism of the underlying trees.…”

A Cantor-Bernstein theorem for infinite matroids

“…A graph-like space Γ is called pseudo-arc-connected if for any u = v ∈ V (Γ) there is a graph-like subspace A of Γ which is pseudo-arc between u and v. Theorem 2.1 (P. J. Gollin and J. Kneip, Theorem 4.6 in [13]). A compact graph-like space is (topologically) connected if and only if it is pseudo-arc connected.…”

“…A graph-like tree is a connected graph-like space without pseudo-circles. Fact 2.3 (Proposition 4.9 of [13]). A compact loop-free graph-like space is a graph-like-tree if and only if each vertex pair is connected by a unique pseudo-arc.…”

Applications of Order Trees in Infinite Graphs

“…As a case in point, recall that in the framework of end spaces, establishing the existence of a nested family of clopen sets witnessing that end spaces are Hausdorff is a deep result by Carmesin [3,Corollary 5.17]. See [9,15] for further applications of nested sets of separations of finite and infinite graphs. In the framework of ray spaces, the definition R(T ) ⊆ 2 T makes it clear that the topology of R(T ) is generated by the subbase obtained by declaring all sets of the form [t] = {x ∈ R(T ) : t ∈ x} to be clopen.…”