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 proved 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 {\sl manifold-like} generalized \(n\)-manifolds \(X^{n},\) introduced in 1966 by Mardeši\'{c} and Segal, which are characterized by the existence of \(\delta\)-mappings \(f_{\delta}\) of \(X^n\) onto closed manifolds \(M^{n}_{\delta},\) for arbitrary small \(\delta>0\), i.e. there exist onto maps \(f_{\delta}\colon X^{n}\to M^{n}_{\delta}\) such that for every \(u\in M^{n}_{\delta}\), \(f^{-1}_{\delta}(u)\) has diameter less than \(\delta\). We prove that with respect to the metric \(d_G,\) manifold-like generalized \(n\)-manifolds \(X^{n}\) are limits of topological \(n\)-manifolds \(M^{n}_{i}\). Moreover, if topological \(n\)-manifolds \(M^{n}_{i}\) satisfy a certain local contractibility condition \(\mathcal{M}(\varrho, n)\), we prove that generalized \(n\)-manifold \(X^{n}\) is resolvable.