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A Continuous Extension Operator for Convex Metrics

Published online by Cambridge University Press:  20 November 2018

I. Stasyuk  and
E. D. Tymchatyn
I. Stasyuk
Affiliation:
Department of Mechanics and Mathematics, Ivan Franko National University of Lviv, Lviv, Ukraine e-mail: i stasyuk@yahoo.com
E. D. Tymchatyn
Affiliation:
Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, SK e-mail: tymchat@math.usask.ca
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Abstract

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We consider the problem of simultaneous extension of continuous convex metrics defined on subcontinua of a Peano continuum. We prove that there is an extension operator for convex metrics that is continuous with respect to the uniform topology.

Keywords

54E3554C2054E40
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 53 , Issue 4 , 01 December 2010 , pp. 719 - 729
Copyright
Copyright © Canadian Mathematical Society 2010

References

[1] Banakh, T., AE(0)-spaces and regular operators extending (averaging) pseudometrics. Bull. Polish Acad. Sci. Math. 42(1994), no. 3, 197–206.Google Scholar
[2] Banakh, T. and Bessaga, C., On linear operators extending [pseudo]metrics. Bull. Polish Acad. Sci. Math. 48(2000), no. 1, 35–49.Google Scholar
[3] Banakh, T., Brodskiy, N., Stasyuk, I., and Tymchatyn, E. D., On continuous extension of uniformly continuous functions and metrics. Colloq. Math. 116(2009), no. 2, 191–202.Google Scholar
[4] Bessaga, C., On linear operators and functors extending pseudometrics. Fund. Math. 142(1993), no. 2, 101–122.Google Scholar
[5] Bing, R. H., A convex metric for a locally connected continuum. Bull. Amer. Math. Soc. 55(1949), 812–819. doi:10.1090/S0002-9904-1949-09298-4Google Scholar
[6] Bing, R. H., Partitioning continuous curves. Bull. Amer. Math. Soc. 58(1952), 536–556. doi:10.1090/S0002-9904-1952-09621-XGoogle Scholar
[7] Hewitt, E. and Stromberg, K., Real and abstract analysis. A modern treatment of the theory of functions of a real variable. Graduate Texts in Mathematics, 25, Springer-Verlag, New York-Heidelberg, 1975.Google Scholar
[8] Moise, E. E., Grille decomposition and convexification theorems for compact metric locally connected continua. Bull. Amer. Math. Soc. 55(1949), 1111–1121. doi:10.1090/S0002-9904-1949-09336-9Google Scholar
[9] Moise, E. E., A note of correction. Proc. Amer. Math. Soc. 2(1951), 838. doi:10.2307/2032089Google Scholar
[10] Stasyuk, I., Operators of simultaneous extensions of partial ultrametrics. (Ukrainian) Mat. Metodi Fiz.-Mekh. Polya 49(2006), no. 2, 27–32.Google Scholar
[11] Stasyuk, I. and Tymchatyn, E. D., A continuous operator extending ultrametrics. Comment. Math. Univ. Carolin. 50(2009), no. 1, 141–151.Google Scholar
[12] Stepanova, E. N., Continuation of continuous functions and the metrizability of paracompact p-spaces. (Russian) Mat. Zametki 53(1993), no. 3, 92–101; translation in Math. Notes 53(1993), no. 3-4, 308–314.Google Scholar
[13] Tymchatyn, E. D. and Zarichnyi, M., On simultaneous linear extensions of partial (pseudo)metrics. Proc. Amer. Math. Soc. 132(2004), no. 9, 2799–2807. doi:10.1090/S0002-9939-04-07413-1Google Scholar
[14] Tymchatyn, E. D. and Zarichnyi, M., A note on operators extending partial ultrametrics. Comment. Math. Univ. Carolin. 46(2005), no. 3, 515–524.Google Scholar