Peano compactifications and property S metric spaces
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-kn...
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Wiley
1980-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S016117128000049X |
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| _version_ | 1832551294159552512 |
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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). There are easily constructed examples of locally connected connected metric spaces which fail to be S-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), X must be S-metrizable. Several (new) necessary and sufficient for a space to be S-metrizable are given, together with an example of non-S-metrizable space which fails to have a Peano compactification. |
| format | Article |
| id | doaj-art-ef626761d30746bdb81c8e90db8d50c1 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1980-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-ef626761d30746bdb81c8e90db8d50c12025-02-03T06:01:49ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251980-01-013469570010.1155/S016117128000049XPeano compactifications and property S metric spacesR. 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). There are easily constructed examples of locally connected connected metric spaces which fail to be S-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), X must be S-metrizable. Several (new) necessary and sufficient for a space to be S-metrizable are given, together with an example of non-S-metrizable space which fails to have a Peano compactification.http://dx.doi.org/10.1155/S016117128000049Xproperty S metricsPeano spacescompactifications. |
| spellingShingle | R. F. Dickman Peano compactifications and property S metric spaces International Journal of Mathematics and Mathematical Sciences property S metrics Peano spaces compactifications. |
| title | Peano compactifications and property S metric spaces |
| title_full | Peano compactifications and property S metric spaces |
| title_fullStr | Peano compactifications and property S metric spaces |
| title_full_unstemmed | Peano compactifications and property S metric spaces |
| title_short | Peano compactifications and property S metric spaces |
| title_sort | peano compactifications and property s metric spaces |
| topic | property S metrics Peano spaces compactifications. |
| url | http://dx.doi.org/10.1155/S016117128000049X |
| work_keys_str_mv | AT rfdickman peanocompactificationsandpropertysmetricspaces |