Some remarks on rectifiably connected metric spaces

We show that the category of intrinsic metric spaces and Lipschitz maps is coreflective in the category of all rectifiably connected metric spaces and Lipschitz maps. We also introduce a class of spaces called rectifiably chain-connected metric spaces, and for such spaces we define the concept of a weakly intrinsic distance. We then prove that the category of rectifiably connected metric spaces with weakly intrinsic distance is coreflective in the category of all rectifiably connected metric spaces and Lipschitz maps. (C) 2018 Elsevier B.V. All rights reserved.