Various topologies on trees

TL;DR

This survey explores various topologies on trees, detailing their properties and relationships, with a focus on wedge and interval topologies, and discusses open problems in the set-theoretic context.

Contribution

It provides a comprehensive overview of different topologies on trees, including detailed analysis of wedge and interval topologies, and highlights open problems in the area.

Findings

Coarse wedge topology yields supercompact monotone normal spaces.

Fine wedge topology makes trees hereditarily ultraparacompact.

Interval topology exhibits diverse properties influenced by set-theoretic axioms.

Abstract

This is a survey article on trees, with a modest number of proofs to give a flavor of the way these topologies can be efficiently handled. Trees are defined in set-theorist fashion as partially ordered sets in which the elements below each element are well-ordered. A number of different topologies on trees are treated, some at considerable length. Two sections deal in some depth with the coarse and fine wedge topologies, and the interval topology, respectively. The coarse wedge topology gives a class of supercompact monotone normal topological spaces, and the fine wedge topology puts a monotone normal, hereditarily ultraparacompact topology on every tree. The interval topology gives a large variety of topological properties, some of which depend upon set-theoretic axioms beyond ZFC. Many of the open problems in this area are given in the last section.