Limits of Manifolds in the Gromov–Hausdorff Metric Space

We apply the Gromov–Hausdorff metric d G for characterization of certain generalized manifolds. Previously, we have proven that with respect to the metric d G , generalized n -manifolds are limits of spaces which are obtained by gluing two topological n -manifolds by a controlled homotopy equivalence (the so-called 2-patch spaces). In the present paper, we consider the so-called manifold-like generalized n -manifolds X n , introduced in 1966 by Mardeić and Segal, which are characterized by the existence of δ -mappings f δ of X n onto closed manifolds M δ n , for arbitrary small δ > 0 , i.e., there exist onto maps f δ : X n → M δ n such that for every u ∈ M δ n , f δ - 1 ( u ) has diameter less than δ . We prove that with respect to the metric d G , manifold-like generalized n -manifolds X n are limits of topological n -manifolds M i n . Moreover, if topological n -manifolds M i n satisfy a certain local contractibility condition M ( ϱ , n ) , we prove that generalized n -manifold X n is resolvable.