Almost convex metrics and Peano compactifications

[摘要] Let(X,d)denote a locally connected, connected separable metric space. We say theXisS-metrizableprovided 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). In an earlier paper, the author conjectured that if a space(X,d)has a Peano compactification, then it must beS-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.

[学科分类] 数学（综合）