Non-separability of the Lipschitz distance

TL;DR

This paper demonstrates that the space of isometry classes of compact metric spaces with finite Lipschitz distance from a fixed space is non-separable when the fixed space is a closed interval or a union of shrinking intervals.

Contribution

It establishes the non-separability of the Lipschitz distance space for specific base metric spaces, revealing fundamental geometric properties.

Findings

$(\\mathcal M_X, d_L)$ is non-separable for $X$ as a closed interval

Non-separability also holds for $X$ as an infinite union of shrinking closed intervals

The result highlights limitations in approximating metric spaces under Lipschitz distance

Abstract

Let $X$ be a compact metric space and $\mathcal M_X$ be the set of isometry classes of compact metric spaces $Y$ such that the Lipschitz distance $d_L(X,Y)$ is finite. We show that $(\mathcal M_X, d_L)$ is not separable when $X$ is a closed interval, or an infinite union of shrinking closed intervals.