Peano compactifications and property S metric spaces

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.