Metric trees in the Gromov--Hausdorff space

Abstract

Using the wedge sum of metric spaces, for all compact metrizable spaces, we construct a topological embedding of the compact metrizable space into the set of all metric trees in the Gromov--Hausdorff space with finite prescribed values. As its application, we show that the set of all metric trees is path-connected and its all non-empty open subsets have infinite topological dimension.

9 pages. This paper is strongly rerated to my papers arXiv:2110.01881 and arXiv:2111.08199. There are some overlapped explanation and definitions