Continua in the Gromov--Hausdorff space

TL;DR

This paper demonstrates that various classes of compact metric spaces, such as connected, path-connected, geodesic, and CAT(0) spaces, are topologically rich and path-connected within the Gromov--Hausdorff space, with infinite topological dimension.

Contribution

It establishes the topological embedding of arbitrary compact metrizable spaces into specific classes of metric spaces within the Gromov--Hausdorff space, revealing their path-connectedness and infinite topological dimension.

Findings

Sets of connected, path-connected, geodesic, and CAT(0) spaces are path-connected.

Non-empty open subsets of these classes have infinite topological dimension.

The set of all proper CAT(0) spaces is also path-connected with infinite dimension.

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.