Contractibility of the space of $\varepsilon$-nets in $\mathbb{R}$

TL;DR

This paper proves that the space of all -nets in the real line, when equipped with Hausdorff or Gromov-Hausdorff metrics, is contractible.

Contribution

It establishes the contractibility of the space of -nets in  under natural metrics, a topological property not previously known.

Findings

The space of -nets in  is contractible.

Contractibility holds under both Hausdorff and Gromov-Hausdorff distances.

Abstract

In this note, we show that the space of all $\varepsilon$-nets in the real line $\mathbb{R}$ with a natural metric, equipped with either Hausdorff or Gromov--Hausdorff distance, is contractible.