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The first two authors introduced the notion of metrically generated theories as a unifying framework for all possible theories which are determined by their metrizable objects (in a general sense, allowing pseudo- quasimetrics). This is notably the case for topological and (quasi-)uniform spaces, however many other examples were provided. We all know that a topology (resp. completely regular topology) can be derived from a quasi- uniformity (resp. uniformity) and that topologies are the structures allowing us to study local concepts, such as continuity whereas uniformities are the structures which allow us to study uniform concepts, such as uniform continuity. We also have formulas showing that indeed, going from a uniformity to the underlying topology involves a kind of localisation of the structure, e.g. by way of deriving neighborhoods from entourages. However all this is fairly meta-mathematical and certainly ad-hoc for the situation of topology and uniformity. In this paper we want to show that, by way of metrically generated theories [4], it is possible to turn these intuitive facts into a general technique. For this, after having recalled the basic notions of metrically generated theories, we will describe what we call local metrically generated theories. We will show that every metrically generated theory gives rise to a unique largest underlying local theory and we will determine precisely what these local underlying theories for various important examples are. In particular we will show that in the case of our penultimate example, namely uniform theories, these largest underlying local theories are indeed the usual topological ones. |
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