A Complete, Bounded and Almost Convex Metric Space which is Not Totally Convex

Carlyle Stewart

A Complete, Bounded and Almost Convex Metric Space which is Not Totally Convex

General Topology 2026-04-10 v1

Authors: Carlyle Stewart

Abstract

In the 2006 paper ``Exponentiation in V-categories'' by M. M. Clementino and D. Hofmann, exponentiable objects in the category of metric spaces with non-expansive maps are characterised as those metric spaces which are bounded and almost convex. A corollary to this characterisation is made claiming that every complete, exponentiable metric space is totally convex. A counterexample to this claim is provided in the present paper, and a correct version of the claim is suggested and proved.

Keywords

metric spaces exponentiable objects almost convex totally convex complete metric spaces counterexample

Cite

@article{arxiv.2607.14094,
  title  = {A Complete, Bounded and Almost Convex Metric Space which is Not Totally Convex},
  author = {Carlyle Stewart},
  journal= {arXiv preprint arXiv:2607.14094},
  year   = {2026}
}

Comments

3 pages, 1 figure