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.
Highlights
For compact metric spaces (X, dX) and (Y, dY ), the Lipschitz distance dL(X, Y ) is defined to be the infimum of ≥ 0 such that an -isometry f : X → Y exists
Let M be the set of isometry classes of compact metric spaces
We change the question (Q) to the following more reasonable one (Q’): For a compact metric space X, let MX be the set of isometry classes of compact metric spaces Y such that dL(X, Y ) < ∞
Summary
1 Introduction For compact metric spaces (X, dX) and (Y , dY ), the Lipschitz distance dL(X, Y ) is defined to be the infimum of ≥ 0 such that an -isometry f : X → Y exists. Let M be the set of isometry classes of compact metric spaces. The following question arises: (Q’): Is the metric space (MX, dL) separable?
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