R. F. Dickman Jr.

Copyright © 1980 R. F. Dickman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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). There are easily constructed examples of locally connected connected metric spaces which fail to be S-metrizable, however the author does not know of a non-S-metrizable space (X,d) which has a Peano compactification. In this paper we conjecture that: If (P,ρ) a Peano compactification of (X,ρ|X), X must be S-metrizable. Several (new) necessary and sufficient for a space to be S-metrizable are given, together with an example of non-S-metrizable space which fails to have a Peano compactification.