AI 中文总结
本文研究Gromov--Hausdorff距离的修改形式，证明$\FF$-距离对态射类Lipschitz连续，并得出光滑距离等于连续距离，且部分连续距离与经典距离一致。
AI 中文摘要
本文研究了Gromov--Hausdorff距离的若干修改形式，其中下确界取遍给定类$\FF$中的态射对——即度量空间范畴的某个子范畴的态射类（称为$\FF$-Gromov--Hausdorff距离）。我们证明了$\FF$-距离在态射上的Hausdorff度量意义下关于类$\FF$是Lipschitz连续的。作为推论，对于配备与拓扑相容的度量的第二可数$C^k$流形，光滑距离与连续距离一致：$\dGH{C_k} = \dGH{C}$。我们还引入了部分连续和部分局部常数距离（连续性被弱化为在满测度支撑的开子集上的连续性），并证明在度量空间类上它们与经典Gromov--Hausdorff距离一致。
英文摘要
This work studies modifications of the Gromov--Hausdorff distance in which the infimum is taken over pairs of morphisms of a given class $\FF$ --- the class of morphisms of a subcategory of the category of metric spaces (the $\FF$-Gromov--Hausdorff distance). We prove that the $\FF$-distance depends Lipschitz-continuously on the class $\FF$ in the Hausdorff metric on morphisms. As a consequence, for second countable $C^k$-manifolds equipped with a metric compatible with the topology, the smooth distance coincides with the continuous one: $\dGH{C_k} = \dGH{C}$. We also introduce the partially continuous and partially locally constant distances (continuity is weakened to continuity on an open subset of the support of full measure) and prove that on the class of metric spaces they coincide with the classical Gromov--Hausdorff distance.
