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More on upper bicompletion-true functorial quasi-uniformities
- Source :
- Topology and its Applications. 158:1937-1941
- Publication Year :
- 2011
- Publisher :
- Elsevier BV, 2011.
-
Abstract
- Let T : QU 0 → Top 0 denote the usual forgetful functor from the category of quasi-uniform T 0 -spaces to that of the topological T 0 -spaces. We regard the bicompletion reflector as a (pointed) endofunctor K : QU 0 → QU 0 . For any section F : Top 0 → QU 0 of T we consider the (pointed) endofunctor R = T K F : Top 0 → Top 0 . 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 Top 0 . An important known characterisation is that F is upper K -true iff the canonical embedding X → R X is an epimorphism in Top 0 for every X in Top 0 . 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 T : QU 0 → Top 0 , 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 .
Details
- ISSN :
- 01668641
- Volume :
- 158
- Database :
- OpenAIRE
- Journal :
- Topology and its Applications
- Accession number :
- edsair.doi.dedup.....39bbee2cf3b823ebc65fcae47eee7ae7