1. More on upper bicompletion-true functorial quasi-uniformities
- Author
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Brümmer, G.C.L., Künzi, Hans-Peter A., and Sioen, M.
- Subjects
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FUNCTOR theory , *UNIFORM spaces , *CATEGORIES (Mathematics) , *MORPHISMS (Mathematics) , *TOPOLOGICAL spaces , *ALGEBRAIC topology , *PROOF theory - Abstract
Abstract: Let denote the usual forgetful functor from the category of quasi-uniform -spaces to that of the topological -spaces. We regard the bicompletion reflector as a (pointed) endofunctor . For any section of T we consider the (pointed) endofunctor . The T-section F is called upper bicompletion-true (briefly, upper K-true) if the quasi-uniform space KFX is finer than FRX for every X in . An important known characterisation is that F is upper K-true iff the canonical embedding is an epimorphism in for every X in . We show that this result admits a simple, purely categorical formulation and proof, independent of the setting of quasi-uniform and topological spaces. We thus mention a few other settings where the result is applicable. Returning then to the setting , we prove: Any T-section F is upper K-true iff for all X the bitopology of KFX equals that of FRX; and iff the join topology of KFX equals the strong topology (also called the b- or Skula topology) of RX. [Copyright &y& Elsevier]
- Published
- 2011
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