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A Topological Characterization of the Length of Paths

In: Selecta Mathematica

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  • Karl Menger

    (Illinois Institute of Technology)

Abstract

Let T be a metrizable topological space. We speak of a universal functional of paths if with every path ß in T a (finite or infinite) number λ ß is associated. For any particular metrization of T the corresponding length of paths is an example of a non-negative universal functional. To different metrizations of T correspond, in general, different lengths. How are these lengths characterized among the non-negative universal functionals of paths? In other words, what properties of a functional λ are necessary and sufficient in order that there exist a metrization of T such that, for every path ß, the corresponding length is equal to λ ß? We widen the scope of the problem by admitting metrizations of T for which the distance is non-symmetric.

Suggested Citation

  • Karl Menger, 2003. "A Topological Characterization of the Length of Paths," Springer Books, in: Bert Schweizer & Abe Sklar & Karl Sigmund & Peter Gruber & Edmund Hlawka & Ludwig Reich & Leopold Sc (ed.), Selecta Mathematica, pages 105-108, Springer.
    • Handle: RePEc:spr:sprchp:978-3-7091-6045-9_16
    DOI: 10.1007/978-3-7091-6045-9_16
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