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Generalized Metric Spaces: Completion, Topology, and Powerdomains via the Yoneda Embedding
, 1996
"... Generalized metric spaces are a common generalization of preorders and ordinary metric spaces (Lawvere 1973). Combining Lawvere's (1973) enrichedcategorical and Smyth' (1988, 1991) topological view on generalized metric spaces, it is shown how to construct 1. completion, 2. topology, and 3. powerdo ..."
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Cited by 23 (3 self)
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Generalized metric spaces are a common generalization of preorders and ordinary metric spaces (Lawvere 1973). Combining Lawvere's (1973) enrichedcategorical and Smyth' (1988, 1991) topological view on generalized metric spaces, it is shown how to construct 1. completion, 2. topology, and 3. powerdomains for generalized metric spaces. Restricted to the special cases of preorders and ordinary metric spaces, these constructions yield, respectively: 1. chain completion and Cauchy completion; 2. the Alexandroff and the Scott topology, and the fflball topology; 3. lower, upper, and convex powerdomains, and the hyperspace of compact subsets. All constructions are formulated in terms of (a metric version of) the Yoneda (1954) embedding.
A "Converse" of the Banach Contraction Mapping Theorem
, 2001
"... this paper that such an ultrametric space underlying these processes can always be found. Thus, our main result, which is stated precisely in Theorem 2, is in a sense a converse of the Banach contraction mapping theorem, and permits that theorem to be "applied" in some circumstances where no metric ..."
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Cited by 1 (1 self)
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this paper that such an ultrametric space underlying these processes can always be found. Thus, our main result, which is stated precisely in Theorem 2, is in a sense a converse of the Banach contraction mapping theorem, and permits that theorem to be "applied" in some circumstances where no metric rendering the operators in question to be contractions was readily to hand in advance
Similarity, Topology, and Uniformity
, 2007
"... 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 ..."
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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 inequation 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 nonexpanding functions modulo some rescaling between different domains of similarity values. We show that nonexpanding functions with locally varying rescaling functions correspond to topologically continuous functions, while nonexpanding functions with a globally fixed rescaling generalize uniformly continuous functions.
Similarity, Topology, and Uniformity Dedicated to Dieter Spreen on the occasion of his 60th birthday
"... 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 ..."
Abstract
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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 nonexpanding functions modulo some rescaling between different domains of similarity values. We show that nonexpanding functions with locally varying rescaling functions correspond to topologically continuous functions, while nonexpanding functions with a globally fixed rescaling generalize uniformly continuous functions.