Noncompactness of segments in the Gromov–Hausdorff space

We study properties of segments in the Gromov–Hausdorff metric space. A segment is a subset of a metric space consisting of points lying between two given points. We prove that any segment in the Gromov–Hausdorff space with endpoints being non-isometric compact metric spaces contains an element that is a compact metric space with at least one isolated point. Using this theorem and Gromov's precompactness criterion, we prove that any nondegenerate segment in the Gromov–Hausdorff space is not a compact set.