Functorial approach structures

Guillaume C.L. Brümmer, M. Sioen

Abstract

We show that there exists at least a proper class of functorial approach structures, i.e., right inverses to the forgetful functor T : AP→ 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 [2] 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 [10], only Top0 itself can be retrieved in this way.


Keywords

Approach space; (approach) bicompleteness; Epireflective subcategory; Functorial approach structure; Spanning; Topological space

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References

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1. Concrete functors determined by their restrictions to the T0 objects
G.C.L. Brümmer, H.-P.A. Künzi, M. Sioen
Topology and its Applications  vol: 158  issue: 17  first page: 2382  year: 2011  
doi: 10.1016/j.topol.2011.03.014



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