International Journal of Mathematics and Mathematical Sciences
Volume 5 (1982), Issue 3, Pages 599-603
Almost convex metrics and Peano compactifications
Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg 24061, Virginia, USA
Received 23 June 1981
Copyright © 1982 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.
Let denote a locally connected, connected separable metric space. We say the is -metrizable provided there is a topologically equivalent metric on such that has Property , i.e., for any , is the union of finitely many connected sets of -diameter less than . It is well-known that -metrizable spaces are locally connected and that if is a Property metric for , then the usual metric completion of is a compact, locally connected, connected metric space; i.e., is a Peano compactification of . In an earlier paper, the author conjectured that if a space has a Peano compactification, then it must be -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.