turkmath.org

Let (X, ρ) be a metric space. The metric ρ is said to be convex if for each distinct x, y ∈ X there exists an arc A ⊂ X with end-points x and y such that A with the restriction of the metric ρ is isometric to the interval [0, ρ(x, y)] in the real line.
In the late 1940s, Bing proved that every Peano continuum (i.e., a compact, connected and locally connected metric space) admits a compatible convex metric and asked for an extension of his theorem to non-compact spaces. In this study, we provide an extension of Bing’s theorem to a quite large class of metric spaces. We prove that every connected and locally arc-connected space with property S admits a convex metric compatible with its topology. A metric space (X, ρ) has property S if for each t > 0 there is a finite cover of X by connected sets of diameter less than t.
This is a joint work with J. Nikiel (University of Opole, Poland), I. Stasyuk (Thompson River University, Canada), and E.D. Tymchatyn (University of Saskatchewan, Canada).