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On covering properties of end and ray spaces

Rodrigo Rey Carvalho, Matheus Duzi, Vinicius de Oliveira Rodrigues

TL;DR

This work analyzes covering properties of end spaces, ray spaces, and edge-end spaces arising from infinite graphs, linking topological questions to underlying graph and tree combinatorics. Using the Kurkofka–Pitz representation that identifies end spaces with ray spaces of rooted trees, the authors develop combinatorial criteria for Lindelöf degree, Rothberger, Menger, and σ-compactness, and prove that these spaces are D-spaces. A central contribution is a construction of rayless normal trees to control unbounded components, yielding precise Lindelöf-degree characterizations and enabling D-space conclusions. The results establish Menger ⇔ σ-compact in these spaces and provide Rothberger characterizations via the absence of Cantor or binary-tree substructures, with analogous extensions to edge-end spaces, thereby enriching the link between graph structure and topology.

Abstract

We provide new results on combinatorial characterizations of covering properties in end spaces and ray spaces. In particular, we characterize the Lindelöf degree, the extent, the Rothberger property, $σ$-compactness and the Menger property for ray, end and edge-end spaces. We show that $σ$-compactness and the Menger property are equivalent for these spaces, and that they are all $D$-spaces.

On covering properties of end and ray spaces

TL;DR

This work analyzes covering properties of end spaces, ray spaces, and edge-end spaces arising from infinite graphs, linking topological questions to underlying graph and tree combinatorics. Using the Kurkofka–Pitz representation that identifies end spaces with ray spaces of rooted trees, the authors develop combinatorial criteria for Lindelöf degree, Rothberger, Menger, and σ-compactness, and prove that these spaces are D-spaces. A central contribution is a construction of rayless normal trees to control unbounded components, yielding precise Lindelöf-degree characterizations and enabling D-space conclusions. The results establish Menger ⇔ σ-compact in these spaces and provide Rothberger characterizations via the absence of Cantor or binary-tree substructures, with analogous extensions to edge-end spaces, thereby enriching the link between graph structure and topology.

Abstract

We provide new results on combinatorial characterizations of covering properties in end spaces and ray spaces. In particular, we characterize the Lindelöf degree, the extent, the Rothberger property, -compactness and the Menger property for ray, end and edge-end spaces. We show that -compactness and the Menger property are equivalent for these spaces, and that they are all -spaces.

Paper Structure

This paper contains 13 sections, 41 theorems, 18 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Basic definitions
  3. On graphs and ends
  4. On trees and rays
  5. Structure of this paper
  6. On the Lindelöf property
  7. Proof of Lemma \ref{['lemma:mainLemma']}:
  8. Proof of Theorem \ref{['theorem:lindelofDegree']}:
  9. Ray spaces are D-spaces
  10. On the Rothberger property and scattered ray spaces
  11. On the Menger property
  12. On edge-end spaces
  13. Final remarks

Key Result

Lemma 1.7

Let $(T, r)$ be a normal tree of a graph $G$. Then for all distinct points $x, y \in T$ which are incompatible in the tree order of $T$, then $\mathring{\left\lceil {x} \right\rceil}\cap \mathring{\left\lceil {y} \right\rceil}$ separates $x$ and $y$ in $G$.

Figures (2)

  • Figure 1: One point compactification of an uncountable discrete space as an end-space.
  • Figure 2: Figure of the graph in \ref{['EX_NotRothEdgeEnds']}, in which no vertex is timid but whose edge-end space is not Rothberger.

Theorems & Definitions (82)

  • Definition 1.1: Graph
  • Definition 1.2: Subgraphs and induced subgraph
  • Definition 1.3: Paths, rays and tails
  • Definition 1.4: Connected graphs, components and trees
  • Definition 1.5: Tree order
  • Definition 1.6: Normal tree
  • Lemma 1.7
  • Definition 1.8: End space
  • Definition 1.9
  • Definition 1.10
  • ...and 72 more