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What does it mean for a shape to change continuously? Over the space ofconvex regions, there is only one "reasonable" answer. However, over a broaderclass of regions, such as the class of star-shaped regions, there can be manydifferent "reasonable" definitions of continuous shape change.  We consider the relation between topologies induced by a number of metricsover a number of limited categories of open bounded regions in n-dimensionalEuclidean space. Specifically, we consider a homeomorphism-based metric; theHausdorff metric; the dual-Hausdorff metric; the symmetric difference metric;and the family of Wasserstein metrics; and the topologies that they induce overthe space of convex regions; the space of convex regions and unions of twoseparated convex regions; and the space of star-shaped regions. We demonstratethat:  Over the space of convex regions, all five metrics, and indeed any metricthat satisfies two general well-behavedness constraints, induce the sametopology.  Over the space of convex regions and unions of two separated convex regions,these five metrics are all ordered by "strictly finer than" relations. Indescending order of fineness, these are: the homeomorphism-based, thedual-Hausdorff, the Hausdorff, the Wasserstein, and the symmetric difference.Also, Wasserstein metrics are strictly ordered among themselves.  Over the space of star-shaped regions, the topologies induced by theHausdorff metric, the symmetric-difference metric, and the Wasserstein metricsare incomparable in terms of fineness.

Metric topologies over some categories of simple open regions in Euclidean space