# Extending pairs of metrics

Author I. Z. Stasyuk, E. D. Tymchatyn
Lviv National University, University of Saskatchewan

Abstract We consider the problem of extension of pairs of continuous and bounded, partial metrics which agree on the non-empty intersections of their domains which are closed and bounded subsets of an arbitrary but fixed metric space. Two pairs of such metrics are close if their corresponding graphs are close and if the intersections of their domains are close in the Hausdorff metric. If, besides, these metrics are uniformly continuous on the intersections of their domains then there is a continuous positive homogeneous operator extending each such a pair of partial metrics to a continuous metric on the union of their domains. We prove that, in general, there is no subadditive extension operator (continuous or not) for such pairs of metrics. We provide examples showing to what extent our results are sharp and we obtain analogous results for ultrametrics.
Keywords Hausdorff metric; ultrametric
Reference 1. T. Banakh, AE(0)–spaces and regular operators extending (averaging) pseudometrics, Bull. Polish Acad. Sci. Math. 42 (1994), ¹3, 197–206.

2. T. Banakh, N. Brodskiy, I. Stasyuk, E. Tymchatyn, On continuous extension of uniformly continuous functions and metrics, Colloq. Math. 116 (2009), 191–202.

3. F. Hausdorff, Erweiterung einer Homomorphie, Fund. Math. 16 (1930), 353–360.

4. R. Bing, Extending a metric, Duke Mathematical Journal 14 (1947), 511–519.

5. J. de Groot, Some special metrics in general topology, Coll. Math. 6 (1958), 283–286.

6. I. Stasyuk, E. Tymchatyn, A continuous operator extending ultrametrics, Comment. Math. Univ. Carolin. 50 (2009), ¹1, 141–151.

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

8. E. Tymchatyn, M. Zarichnyi, A note on operators extending partial ultrametrics, Comment. Math. Univ. Carolinae 46 (2005), ¹3, 515–524.

9. Ñ. Áîãàòûé, Ìåòðè÷åñêè îäíîðîäíûå ïðîñòðàíñòâà, Óñïåõè ìàò. íàóê 57 (2002), ¹2(344), 2–22.

Pages 215-224
Volume 35
Issue 2
Year 2011
Journal Matematychni Studii
Full text of paper PDF
Table of content of issue HTML