Generic Compacta from Relations between Finite Graphs: Theory Building and Examples

TL;DR

This paper explores how finite graphs and their relations can be used to construct topological spaces, linking combinatorial graph properties with topological features, and demonstrates how well-known continua can be built as limits of finite graphs.

Contribution

It introduces a method to construct topological spaces from finite graphs and relations, connecting combinatorial properties with topological spectra, and shows how to realize continua as Fraïssé limits of finite graphs.

Findings

Topological properties relate to graph category properties

Elementary constructions of continua as Fraïssé limits

Method bridges combinatorics and topology

Abstract

In recent work, the authors developed a simple method of constructing topological spaces from certain well-behaved partially ordered sets -- those coming from sequences of relations between finite sets. This method associates a given poset with its spectrum, which is a compact T_1 topological space. In this paper, we focus on the case where such finite sets have a graph structure and the relations belong to a given graph category. We relate topological properties of the spectrum to combinatorial properties of the graph categories involved. We then utilise this to exhibit elementary combinatorial constructions of well-known continua as Fra\"iss\'e limits of finite graphs in categories with relational morphisms.