Topological groups from matrices of sets

TL;DR

This paper introduces a method to construct topological groups from combinatorial structures like trees and functions, linking their properties and set-theoretic assumptions to topological characteristics.

Contribution

It provides a general framework for building topological groups from combinatorial objects and relates their properties to the underlying structures and set-theoretic conditions.

Findings

Groups from ω₁-trees are Frechet iff the tree is Aronszajn.

Cofinal types of these groups are determined under specific set-theoretic assumptions.

Connections established between combinatorial properties and topological group characteristics.

Abstract

We give a general construction of topological groups from combinatorial structures such as trees, towers, gaps, and subadditive functions. We connect topological properties of corresponding groups with combinatorial properties of these objects. For example, the group built from an $\omega_1$-tree is Frech\'{e}t iff the tree is Aronszajn. We also determine cofinal types of some of these groups under certain set theoretic assumptions.