Similarity, topology, and uniformity

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Abstract

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 are 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 rescaling generalize uniformly continuous functions. © 2009 Elsevier Inc. All rights reserved.

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Heckmann, R. (2010). Similarity, topology, and uniformity. Journal of Logic and Algebraic Programming, 79(1), 10–31. https://doi.org/10.1016/j.jlap.2009.02.003

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