  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.
