On a homogeneous operator extending partial ultrametrics

I.Z. Stasyuk

Recently, E.D. Tymchatyn and M.M. Zarichnyi constructed a continuous extension operator for partial ultrametrics on a zero-dimensional compact metrizable space. This extension operator preserves the maximum of two ultrametrics but fails to preserve the homogeneity. The aim of this note is to provide a continuous homogeneous operator that extends partial ultrametrics and preserves norms.

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

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

4. Banakh T., Tymchatyn E.D., Zarichnyi M. Extensions of metrics: survey of results, (in preparation).

5. Tymchatyn E.D., Zarichnyi M. On simultaneous linear extensions of partial (pseudo)metrics, Proc. Amer. Math. Soc. 132 (2004), 2799–2807.

6. Mazurkiewicz S. Sur l'espace des continus peaniens, Fund. Math. 24 (1935), 118–134.

7. Tymchatyn E.D., Zarichnyi M. A note on operators extending partial ultrametrics, Preprint.

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

9. Nadler S.B. Hyperspaces of sets, Marcel Dekker, Inc., New York and Basel, 1978.

10. Assouad P. Sur la distance de Nagata. C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), no.1, 31–34.

11. Movahedi-Lankarani H. An invariant of bi-Lipschitz maps, Fund. Math. 143 (1993), no.1, 1–9.

	On a homogeneous operator extending partial ultrametrics
Author	I. Z. Stasyuk topology@franko.lviv.ua Lviv Ivan Franko National University
Abstract	Recently, E.D. Tymchatyn and M.M. Zarichnyi constructed a continuous extension operator for partial ultrametrics on a zero-dimensional compact metrizable space. This extension operator preserves the maximum of two ultrametrics but fails to preserve the homogeneity. The aim of this note is to provide a continuous homogeneous operator that extends partial ultrametrics and preserves norms.
Keywords	continuous extension operator, partial ultrametric, zero-dimensional compact metrizable space
DOI	doi:10.30970/ms.22.1.73-78
Reference	1. Hausdorff F. Erweiterung einer Homöomorphie, Fund. Math. 16 (1930), 353–360. 2. Bessaga C. On linear operators and functors extending pseudometrics, Fund. Math. 142 (1993), no.2, 101–122. 3. Banakh T. $\rm AE(0)$-spaces and regular operators extending (averaging) pseudometrics, Bull. Polish Acad. Sci. Math 42 (1994), no.3, 197--206. 4. Banakh T., Tymchatyn E.D., Zarichnyi M. Extensions of metrics: survey of results, (in preparation). 5. Tymchatyn E.D., Zarichnyi M. On simultaneous linear extensions of partial (pseudo)metrics, Proc. Amer. Math. Soc. 132 (2004), 2799–2807. 6. Mazurkiewicz S. Sur l'espace des continus peaniens, Fund. Math. 24 (1935), 118–134. 7. Tymchatyn E.D., Zarichnyi M. A note on operators extending partial ultrametrics, Preprint. 8. de Groot J. Non-archimedean metrics in topology, Proc. Amer. Math. Soc. 7 (1956), 948–953. 9. Nadler S.B. Hyperspaces of sets, Marcel Dekker, Inc., New York and Basel, 1978. 10. Assouad P. Sur la distance de Nagata. C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), no.1, 31–34. 11. Movahedi-Lankarani H. An invariant of bi-Lipschitz maps, Fund. Math. 143 (1993), no.1, 1–9.
Pages	73-78
Volume	22
Issue	1
Year	2004
Journal	Matematychni Studii
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