On the Gromov-Hausdorff distance between compact spaces

Abstract

This work provides an introduction to the Gromov-Hausdorff distance, discussing its original definition and its relationship with correspondences between spaces. We prove that the Gromov-Hausdorff distance serves as a metric for the set of isometry classes of compact metric spaces. The primary objectives of this study are to establish the existence of a pseudo-metric on the disjoint union $X \sqcup Y$ that achieves the Gromov-Hausdorff distance between compact spaces $X$ and $Y$, and to establish bounds for the Gromov-Hausdorff distance between spheres of different dimensions.