Functorial approach structures
Keywords:Approach space, (approach) bicompleteness, Epireflective subcategory, Functorial approach structure, Spanning, Topological space
We show that there exists at least a proper class of functorial approach structures, i.e., right inverses to the forgetful functor T : APâ†’ Top (where AP denotes the topological construct of approach spaces and contractions as introduced by R. Lowen). There is however a great difference in nature of these functorial approach structures when compared to the quasi-uniform paradigm which has been extensively studied by the first author: whereas it is well-known from  that a large class of epireflective subcategories of Top0 can be “parametrized” using the interaction of functorial quasi-uniformities with the quasi-uniform bicompletion, we show that using functorial approach structures together with the approach bicompletion developed in , only Top0 itself can be retrieved in this way.
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