Almost convex metrics and Peano compactifications - R. F. Dickman (International Journal of Mathematics and Mathematical Sciences)
Let (X,d) denote a locally connected, connected separable metric space. We say the X is S-metrizable provided there is a topologically equivalent metric ρ on X such that (X,ρ) has Property S, i.e., for any ϵ>0, X is the union of finitely many connected sets of ρ-diameter less than ϵ. It is well-known that S-metrizable spaces are locally connected and that if ρ is a Property S metric for X, then the usual metric completion (X˜,ρ˜) of (X,ρ) is a compact, locally connected, connected metric space; i.e., (X˜,ρ˜) is a Peano compactification of (X,ρ). In an earlier paper, the author conjectured that if a space (X,d) has a Peano compactification, then it must be S-metrizable. In this paper, that conjecture is shown to be false; however, the connected spaces which have Peano compactificatons are shown to be exactly those having a totally bounded, almost convex metric. Several related results are given.