In this paper, we first evaluate topological distributions of the sets of all doubling spaces, all uniformly disconnected spaces, and all uniformly perfect spaces in the space of all isometry classes of compact metric spaces equipped with the Gromov–Hausdorff distance.We then construct branching geodesics of the Gromov–Hausdorff distance continuously parameterized by the Hilbert cube, passing through or avoiding sets of all spaces satisfying some of the three properties shown above, and passing through the sets of all infinite-dimensional spaces and the set of all Cantor metric spaces. Our construction implies that for every pair of compact metric spaces, there exists a topological embedding of the Hilbert cube into the Gromov– Hausdorff space whose image contains the pair. From our results, we observe that the sets explained above are geodesic spaces and infinite-dimensional.

中文翻译：

---

Gromov-Hausdorff 距离的分支测地线

在本文中，我们首先评估了配备 Gromov-Hausdorff 距离的紧度量空间的所有等距类空间中所有加倍空间、所有一致断开空间和所有一致完美空间的集合的拓扑分布。然后我们构造分支Gromov-Hausdorff 距离的测地线由希尔伯特立方体连续参数化，通过或避免满足上述三个属性中的一些的所有空间的集合，并通过所有无限维空间的集合和所有康托尔度量空间的集合. 我们的构造意味着对于每一对紧凑度量空间，都存在将希尔伯特立方体拓扑嵌入到图像包含该对的格罗莫夫-豪斯多夫空间中。从我们的结果来看，