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 all its nonempty open subsets have infinite topological dimension.
Keywords: metric tree; Gromov-Hausdorff distance
DOI: DOI 10.14712/1213-7243.2023.012
AMS Subject Classification: 53C23 51F99