Similarity, topology, and uniformity

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Similarity, topology, and uniformity
We generalize various notions of generalized metrics even further to one general concept comprising them all. For convenience, we turn around the ordering in the target domain of the generalized metrics so that we speak of similarity instead of distance. Starting from an extremely general situation without axioms, we examine which axioms or additional properties are needed to obtain useful results. For instance, we shall see that commutativity and associativity of the generalized version of addition occurring in the triangle inequality is not really needed, nor do we require a generalized version of subtraction. Each similarity space comes with its own domain of possible similarity values. Therefore, we consider non-expanding functions modulo some rescaling between different domains of similarity values. We show that non-expanding functions with locally varying rescaling functions correspond to topologically continuous functions, while non-expanding functions with a globally fixed r...
Reinhold Heckmann
Added 28 Jan 2011
Updated 28 Jan 2011
Type Journal
Year 2010
Where JLP
Authors Reinhold Heckmann
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