On convergence structures in infinite graphs

TL;DR

This work develops a net-based convergence framework for infinite graphs by defining a natural convergence on the vertex set through $\,\varphi\to v$ when $N[v]\in \varphi^{\uparrow}$ and adjoining this to a pretopological structure with the closure analogue $dh(A)=N[A]$. It shows that, within this framework, graph-theoretic properties such as connectedness and compactness correspond to convergence-space notions, with connected graphs being path-connected and compactness equivalent to possessing a finite dominating set; it also establishes that the graph convergence is topological exactly for transitive graphs and links graph homomorphisms to continuous maps in the convergence-space category. The paper then develops the topological modification of graphs, analyzes open/closed sets, and demonstrates a clean alignment of products, subspaces, and homomorphisms with the corresponding convergence-space constructions, thereby providing a categorical bridge between graphs and pretopological spaces. Finally, it outlines future directions, including edge-convergence via line graphs, end-space convergences, and refined notions of compactness and domination in infinite graphs, aiming to extend the convergence-closure dictionary to broader combinatorial settings.

Abstract

It is well known that a graph admits a natural closure operator defined on its vertex set, which also corresponds to a pretopology. In this paper, we describe the classical pretopology on a graph in terms of nets, with the aim of relating combinatorial properties to convergence properties.

Figures (6)

• Figure 1: The rays on the left are equivalent, while those on the right do not are.
• Figure 2: A ray converging to a vertex.
• Figure 3: A homotopy between equivalent rays.
• Figure 4: An example of a compact countable graph
• Figure 5: This graph is compact and has infinitely many ends.

Theorems & Definitions (52)

• Theorem 1: \ref{['path']}
• Theorem 2: \ref{['connected']}
• Theorem 3: \ref{['teocomp2']}
• Proposition 1.1: monteiro2024algebraictopologyopensets
• Lemma 1.1: Pasting lemma nettopology
• Theorem 2.1
• proof
• Proposition 2.1
• proof
• Proposition 2.2