Abstract
This chapter discusses the theory of Hausdorff convergence. Classically, the Hausdorff distance between two closed subsets in a fixed metric space has been defined. Gromov defined the Hausdorff distance between two metric spaces by taking the infimum of the Hausdorff distances over all ambient spaces into which the two metric spaces are embedded by isometries. There he proved two fundamental results—the precompactness theorem and the convergence theorem—and applied them to the study of the global structures of Riemannian manifolds. The chapter also discusses generalizations of Hausdorff convergence to the case of open manifolds and/or manifolds on which groups act.
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