On operators extending uniformly disconnected metrics

I.Z.Stasyuk

We consider the problem of simultaneous extension of continuous uniformly disconnected metrics defined on compact subsets of a zero-dimensional metrizable compact topological space. We prove that the construction of operators extending partial ultrametrics from recent publications of E. D. Tymchatyn, M. M. Zarichnyi and the author can be applied to extending uniformly disconnected metrics.

1. Hausdorff F. Erweiterung einer Hom\"oomorphie, Fund. Math. 16 (1930), 353--360.

2. Bessaga C. Functional analytic aspects of geometry. Linear extending of metrics and related problems. Progress in functional analysis (Pe$\widetilde{\rm n}$\' scola, 1990), North-Holland Math. Stud. 170 (1992), North-Holland, Amsterdam, 247--257.

3. Bessaga C. On linear operators and functors extending pseudometrics, Fund. Math. 142 (1993), no.2, 101--122.

4. Banakh T. $\rm AE(0)$--spaces and regular operators extending (averaging) pseudometrics, Bull. Polish Acad. Sci. Math. 42 (1994), no.3, 197--206.

5. Banakh T., Bessaga C. On linear operators extending [pseudo]metrics, Bull. Polish Acad. Sci. Math. 48 (2000), no.1, 35--49.

6. Zarichnyi M. Regular linear operators extending metrics: a short proof, Bull. Pol. Acad. Sci. Math. 44 (1996), no.3, 267--269.

7. Tymchatyn E. D., Zarichnyi M. A note on operators extending partial ultrametrics, Comment. Math. Univ. Carolinae 46 (2005), no.3, 515--524.

8. Стасюк І. З. Оператори одночасного продовження часткових ультраметрик (подано у Мат. методи і фіз.-мех. поля).

9. Stasyuk I. Z. On a homogeneous operator extending partial ultrametrics, Mat. Stud. 22 (2004), no.1, 73--78.

10. Assouad P. Sur la distance de Nagata, C. R. Acad. Sci. Paris S\'er. I Math 294 (1982), no.1, 31--34.

11. Assouad P. Plongements lipschitziens dans ${\Bbb R}^n$, Bull. Soc. Math. France 111 (1983), 429--448.

12. David G., Semmes S. Fractured fractals and broken dreams: Self-similar geometry through metric and measure, Oxford Lecture Ser. Math. Appl. 7, Oxford University Press, 1997.

13. de Groot J. Non-archimedean metrics in topology, Proc. Amer. Math. Soc. 7 (1956), 948--953.

	On operators extending uniformly disconnected metrics
Author	I.Z.Stasyuk i_stasyuk@yahoo.com Lviv Ivan Franko National University
Abstract	We consider the problem of simultaneous extension of continuous uniformly disconnected metrics defined on compact subsets of a zero-dimensional metrizable compact topological space. We prove that the construction of operators extending partial ultrametrics from recent publications of E. D. Tymchatyn, M. M. Zarichnyi and the author can be applied to extending uniformly disconnected metrics.
Keywords	continuous uniformly disconnected metric, zero-dimensional metrizable compact topological space, partial ultrametric
DOI	doi:10.30970/ms.26.1.101-104
Reference	1. Hausdorff F. Erweiterung einer Hom\"oomorphie, Fund. Math. 16 (1930), 353--360. 2. Bessaga C. Functional analytic aspects of geometry. Linear extending of metrics and related problems. Progress in functional analysis (Pe$\widetilde{\rm n}$\' scola, 1990), North-Holland Math. Stud. 170 (1992), North-Holland, Amsterdam, 247--257. 3. Bessaga C. On linear operators and functors extending pseudometrics, Fund. Math. 142 (1993), no.2, 101--122. 4. Banakh T. $\rm AE(0)$--spaces and regular operators extending (averaging) pseudometrics, Bull. Polish Acad. Sci. Math. 42 (1994), no.3, 197--206. 5. Banakh T., Bessaga C. On linear operators extending [pseudo]metrics, Bull. Polish Acad. Sci. Math. 48 (2000), no.1, 35--49. 6. Zarichnyi M. Regular linear operators extending metrics: a short proof, Bull. Pol. Acad. Sci. Math. 44 (1996), no.3, 267--269. 7. Tymchatyn E. D., Zarichnyi M. A note on operators extending partial ultrametrics, Comment. Math. Univ. Carolinae 46 (2005), no.3, 515--524. 8. Стасюк І. З. Оператори одночасного продовження часткових ультраметрик (подано у Мат. методи і фіз.-мех. поля). 9. Stasyuk I. Z. On a homogeneous operator extending partial ultrametrics, Mat. Stud. 22 (2004), no.1, 73--78. 10. Assouad P. Sur la distance de Nagata, C. R. Acad. Sci. Paris S\'er. I Math 294 (1982), no.1, 31--34. 11. Assouad P. Plongements lipschitziens dans ${\Bbb R}^n$, Bull. Soc. Math. France 111 (1983), 429--448. 12. David G., Semmes S. Fractured fractals and broken dreams: Self-similar geometry through metric and measure, Oxford Lecture Ser. Math. Appl. 7, Oxford University Press, 1997. 13. de Groot J. Non-archimedean metrics in topology, Proc. Amer. Math. Soc. 7 (1956), 948--953.
Pages	101-104
Volume	26
Issue	1
Year	2006
Journal	Matematychni Studii
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