Metric trees in the Gromov--Hausdorff space

TL;DR

This paper constructs a topological embedding of any compact metrizable space into the space of metric trees within the Gromov--Hausdorff space, revealing properties like path-connectedness and infinite topological dimension of the set of all metric trees.

Contribution

It introduces a method to embed any compact metrizable space into metric trees in the Gromov--Hausdorff space, demonstrating new topological properties of this set.

Findings

The set of all metric trees in the Gromov--Hausdorff space is path-connected.

All non-empty open subsets of this set have infinite topological dimension.

A new embedding technique using wedge sums of metric spaces was developed.

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.