Gromov-Hausdorff limits in definable families

Abstract

The notion of piecewise definable metric space is introduced. It is shown that the set of all Gromov-Hausdorff limits of piecewise definable spaces belonging to a fixed bounded definable family is again a definable family. The Gromov-Hausdorff limit is taken with respect to the geodesic metric and the word definable means definable in some o-minimal structure over $\mathbb{R}$.