Almost convex metrics and Peano compactifications
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-...
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| Language: | English |
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Wiley
1982-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171282000568 |
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| _version_ | 1832556011488018432 |
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| author | R. F. Dickman |
| author_facet | R. F. Dickman |
| author_sort | R. F. Dickman |
| collection | DOAJ |
| description | 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). In an earlier paper, the author conjectured that if a space (X,d) has a Peano compactification, then it must be S-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. |
| format | Article |
| id | doaj-art-a8af5973cedb45419b80ecaa5f338ee2 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1982-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-a8af5973cedb45419b80ecaa5f338ee22025-02-03T05:46:32ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251982-01-015359960310.1155/S0161171282000568Almost convex metrics and Peano compactificationsR. F. Dickman0Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg 24061, Virginia, USALet (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). In an earlier paper, the author conjectured that if a space (X,d) has a Peano compactification, then it must be S-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.http://dx.doi.org/10.1155/S0161171282000568almost convex metricsproperty S metricsPeano spacescompactifications. |
| spellingShingle | R. F. Dickman Almost convex metrics and Peano compactifications International Journal of Mathematics and Mathematical Sciences almost convex metrics property S metrics Peano spaces compactifications. |
| title | Almost convex metrics and Peano compactifications |
| title_full | Almost convex metrics and Peano compactifications |
| title_fullStr | Almost convex metrics and Peano compactifications |
| title_full_unstemmed | Almost convex metrics and Peano compactifications |
| title_short | Almost convex metrics and Peano compactifications |
| title_sort | almost convex metrics and peano compactifications |
| topic | almost convex metrics property S metrics Peano spaces compactifications. |
| url | http://dx.doi.org/10.1155/S0161171282000568 |
| work_keys_str_mv | AT rfdickman almostconvexmetricsandpeanocompactifications |