Some remarks on rectifiably connected metric spaces
Proceeding with the studies of previous articles on connectedness in metric spaces, the authors provide, through constructive proofs and using categorical language, coreflectiveness theorems. Namely, first considering Lipschitz maps as morphisms, the category of intrinsic metric spaces (or, equivalently, length spaces) is shown to be coreflective in the category of rectifiably connected metric spaces (or, equivalently, Lipschitz connected metric spaces). Changing from Lipschitz maps to contractive maps, the latter result is still valid. However, the question of coreflectiveness remains open when considering (uniformly) continuous maps as the morphisms, and a counter-example is presented proving that such coreflection cannot be given in a trivial way. Adding a finiteness condition to the concept of chain-connectedness, the class of rectifiably chain-connected metric spaces is introduced, leading to the definition of weakly intrinsic metrics. Every rectifiably connected metric space is proven to be rectifiably chain-connected; moreover, the category of rectifiably connected metric spaces with weakly intrinsic metrics is coreflective in the category of rectifiably chain-connected metric spaces, where the morphisms are considered to be Lipschitz maps, and later on to be contractive maps. Furthermore, counter-examples are provided proving that weakly intrinsic metrics of rectifiably chain-connected metric spaces are not idempotent, and also presenting a rectifiably chain-connected metric space that, when considered with its weakly intrinsic metric, is not even chain-connected. In particular, this proves that a coreflection from the category of rectifiably chain-connected metric spaces with weakly intrinsic metrics to the category of rectifiably chain-connected metric spaces cannot be obtained in a trivial manner, where morphisms are Lipschitz maps. The paper is finished with a result on proper rectifiably connected metric spaces: for such spaces, if the metric is weakly intrinsic, then it must be intrinsic.
scientific article; zbMATH DE number 6986555