Continua in the Gromov--Hausdorff space
Abstract
We first prove that for all compact metrizable spaces, there exists a topological embedding of the compact metrizable space into each of the sets of compact metric spaces which are connected, path-connected, geodesic, or CAT(0), in the Gromov--Hausdorff space with finite prescribed values. As its application, we show that the sets prescribed above are path-connected and their non-empty open subsets have infinite topological dimension. By the same method, we also prove that the set of all proper CAT(0) spaces is path-connected and its non-empty open subsets have infinite topological dimension with respect to the pointed Gromov--Hausdorff distance.
Keywords
Cite
@article{arxiv.2111.08199,
title = {Continua in the Gromov--Hausdorff space},
author = {Yoshito Ishiki},
journal= {arXiv preprint arXiv:2111.08199},
year = {2022}
}
Comments
11 pages. The main results are more generalized. This paper is strongly related to my paper arXiv:2110.01881, and there are overlapped or similar explanations and arguments