| Title:
|
A note on operators extending partial ultrametrics (English) |
| Author:
|
Tymchatyn, E. D. |
| Author:
|
Zarichnyi, M. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
46 |
| Issue:
|
3 |
| Year:
|
2005 |
| Pages:
|
515-524 |
| . |
| Category:
|
math |
| . |
| Summary:
|
We consider the question of simultaneous extension of partial ultrametrics, i.e. continuous ultrametrics defined on nonempty closed subsets of a compact zero-dimensional metrizable space. The main result states that there exists a continuous extension operator that preserves the maximum operation. This extension can also be chosen so that it preserves the Assouad dimension. (English) |
| Keyword:
|
partial ultrametric |
| Keyword:
|
extension operator |
| Keyword:
|
Assouad dimension |
| MSC:
|
54C20 |
| MSC:
|
54E35 |
| MSC:
|
54E40 |
| idZBL:
|
Zbl 1121.54045 |
| idMR:
|
MR2174529 |
| . |
| Date available:
|
2009-05-05T16:52:56Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119545 |
| . |
| Reference:
|
[1] Assouad P.: Sur la distance de Nagata.C.R. Acad. Sci. Paris Sér. I Math. 294 (1982), 1 31-34. Zbl 0481.54015, MR 0651069 |
| Reference:
|
[2] Assouad P.: Plongements lipschitziens dans $\Bbb R^n$.Bull. Soc. Math. France 111 (1983), 429-448. MR 0763553 |
| Reference:
|
[3] Banakh T.: On linear operators extending (pseudo)metrics.preprint. Zbl 0948.54021 |
| Reference:
|
[4] Banakh T.: AE(0)-spaces and regular operators extending (averaging) pseudometrics.Bull. Polish Acad. Sci. Math 42 (1994), 3 197-206. Zbl 0827.54010, MR 1811849 |
| Reference:
|
[5] Banakh T., Bessaga C.: On linear operators extending $[$pseudo$]$metrics.Bull. Polish Acad. Sci. Math. 48 (2000), 1 35-49. Zbl 0948.54021, MR 1751152 |
| Reference:
|
[6] Bessaga C.: On linear operators and functors extending pseudometrics.Fund. Math. 142 (1993), 2 101-122. Zbl 0847.54033, MR 1211761 |
| Reference:
|
[7] Bessaga C.: Functional analytic aspects of geometry. Linear extending of metrics and related problems.Progress in functional analysis (Pe níscola, 1990), 247-257, North-Holland Math. Stud., 170, North-Holland, Amsterdam, 1992. Zbl 0771.54027, MR 1150751 |
| Reference:
|
[8] Engelking R.: General Topology.PWN, Warsaw, 1977. Zbl 0684.54001, MR 0500780 |
| Reference:
|
[9] Filippov V.V.: Topological structure of solution spaces of ordinary differential equations (in Russian).Uspekhi Mat. Nauk 48 (1993), 103-154. MR 1227948 |
| Reference:
|
[10] de Groot J.: Non-archimedean metrics in topology.Proc. Amer. Math. Soc. 7 (1956), 948-953. Zbl 0072.40201, MR 0080905 |
| Reference:
|
[11] Künzi H.-P., Shapiro L.B.: On simultaneous extension of continuous partial functions.Proc. Amer. Math. Soc. 125 (1997), 1853-1859. MR 1415348 |
| Reference:
|
[12] Kuratowski K.: Sur l'espace des fonctions partielles.Ann. Mat. Pura Appl. 40 (1955), 61-67. Zbl 0065.34303, MR 0074807 |
| Reference:
|
[13] Kuratowski K.: Sur une méthode de métrisation complète de certains espaces d'ensembles compacts.Fund. Math. 43 (1956), 114-138. Zbl 0071.38402, MR 0079258 |
| Reference:
|
[14] Luosto K.: Ultrametric spaces bi-Lipschitz embeddable in $\bold R\sp n$.Fund. Math. 150 (1996), 1 25-42. MR 1387955 |
| Reference:
|
[15] Luukkainen J.: Assouad dimension: antifractal metrization, porous sets, and homogeneous measures.J. Korean Math. Soc. 35 (1998), 1 23-76. Zbl 0893.54029, MR 1608518 |
| Reference:
|
[16] Luukkainen J., Movahedi-Lankarani H.: Minimal bi-Lipschitz embedding dimension of ultrametric spaces.Fund. Math. 144 (1994), 2 181-193. Zbl 0807.54025, MR 1273695 |
| Reference:
|
[17] Michael E.: Continuous selections. II.Ann. of Math. (2) 64 (1956), 562-580. Zbl 0073.17702, MR 0080909 |
| Reference:
|
[18] Nadler S.B.: Hyperspaces of Sets.Marcel Dekker, New York and Basel, 1978. Zbl 1125.54001, MR 0500811 |
| Reference:
|
[19] Pikhurko O.: Extending metrics in compact pairs.Mat. Stud. 3 (1994), 103-106, 122. Zbl 0927.54029, MR 1692801 |
| Reference:
|
[20] Stepanova E.N.: Continuation of continuous functions and the metrizability of paracompact $p$-spaces (in Russian).Mat. Zametki 53 (1993), 3 92-101; translation in Math. Notes 53 (1993), no. 3-4, 308-314. MR 1220188 |
| Reference:
|
[21] Tymchatyn E.D., Zarichnyi M.: On simultaneous linear extensions of partial (pseudo)metrics.Proc. Amer. Math. Soc. 132 9 (2004), 2799-2807. Zbl 1050.54011, MR 2054807 |
| Reference:
|
[22] Priess-Crampe S., Ribenboim P.: Generalized ultrametric spaces. I.Abh. Math. Sem. Univ. Hamburg 66 (1996), 55-73. Zbl 0922.54028, MR 1418219 |
| Reference:
|
[23] Zarichnyi M.: Regular linear operators extending metrics: a short proof.Bull. Polish Acad. Sci. Math. 44 (1996), 3 267-269. Zbl 0866.54017, MR 1419399 |
| . |
