International Journal of Mathematics and Mathematical Sciences
Volume 3 (1980), Issue 4, Pages 695-700
Peano compactifications and property metric spaces
Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg 24061, Virginia, USA
Received 17 January 1980
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.
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 . There are easily constructed examples of locally connected connected metric spaces which fail to be -metrizable, however the author does not know of a non--metrizable space which has a Peano compactification. In this paper we conjecture that: If a Peano compactification of , must be -metrizable. Several (new) necessary and sufficient for a space to be -metrizable are given, together with an example of non--metrizable space which fails to have a Peano compactification.