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ON METRIZATION OF UNIONS OF FUNCTION SPACES ON DIFFERENT INTERVALS

Published online by Cambridge University Press:  04 March 2013

TALEB ALKURDI ,
SANDER C. HILLE  and
ONNO VAN GAANS
TALEB ALKURDI*
Affiliation:
Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, The Netherlands
SANDER C. HILLE
Affiliation:
Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, The Netherlands email shille@math.leidenuniv.nl
ONNO VAN GAANS
Affiliation:
Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, The Netherlands email vangaans@math.leidenuniv.nl
*
✉taleb@math.leidenuniv.nl
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Abstract

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This paper investigates a class of metrics that can be introduced on the set consisting of the union of continuous functions defined on different intervals with values in a fixed metric space, where the union ranges over a family of intervals. Its definition is motivated by the Skorohod metric(s) on càdlàg functions. We show what is essential in transferring the ideas employed in the latter metric to our setting and obtain a general construction for metrics in our case. Next, we define the metric space where elements are sequences of functions from the above mentioned set. We provide conditions that ensure separability and completeness of the constructed metric spaces.

Keywords

space of continuous functions on different intervalsmetric spacepenalty functionSkorohod metricconcatenation maplipeomorphism
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 92 , Issue 3 , June 2012 , pp. 281 - 297
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc.

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