Some remarks on rectifiably connected metric spaces
type: Item
instance of: scholarly article
title: Some remarks on rectifiably connected metric spaces
zbMATH Open document ID: 1408.54006
author: Dharmanand Baboolal
author: Paranjothi Pillay
published in: Topology and its Applications
publication date: 2018-11-29T00:00:00Z
review text: 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 …
reviewed by: Willian Ribeiro
Mathematics Subject Classification ID: 54E35
Mathematics Subject Classification ID: 54E40
Mathematics Subject Classification ID: 18A40
zbMATH DE Number: 6986555
zbMATH Keywords: metric space
zbMATH Keywords: Lipschitz map
zbMATH Keywords: length space
zbMATH Keywords: rectifiably chain-connected
zbMATH Keywords: rectifiably connected
zbMATH Keywords: Lipschitz connected
MaRDI profile type: Publication
full work available at URL: j.topol.2018.10.012
OpenAlex ID: W2900485701
cites work: A note on locally pathwise connected metric spaces
cites work: On uniform Lipschitz-connectedness in metric spaces
cites work: Completeness and the Contraction Principle
cites work: Universal locally connected refinements
cites work: Q4394844
cites work: Q5541833
cites work: A course in metric geometry
cites work: Measurability of equivalence classes and MEC\(_p\)-property
cites work: Metric spaces, convexity and nonpositive curvature
Wikidata QID: Q128959463
DOI: 10.1016/J.TOPOL.2018.10.012
description: scientific article; zbMATH DE number 6986555