Typical elements and meager spaces

Let Kn be the set of all compact subsets of Rn, metrized by the Hausdorff metric. We characterize the typical element of the set of all elements of Kn that contain an open ε-ball. In the second section of the paper, we prove several subspaces of Kn consisting of submanifolds of Rn are meager. In the final section of the paper, we provide an elementary proof that the subspace of all compact, connected and locally connected sets is meager. We also prove that the subspace of all compact arc-connected sets is meager.