Peano compactifications and propertySmetric spaces
Let(X,d)denote a locally connected, connected separable metric space. We say theXisS-metrizable provided there is a topologically equivalent metricρonXsuch that(X,ρ)has PropertyS, i.e. for anyϵ>0,Xis the union of finitely many connected sets ofρ-diameter less thanϵ. It is well-known thatS-metrizable spaces are locally connected and that ifρis a PropertySmetric forX, 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 beS-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),Xmust beS-metrizable. Several (new) necessary and sufficient for a space to beS-metrizable are given, together with an example of non-S-metrizable space which fails to have a Peano compactification.