Non-separability of the Lipschitz distance

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.