Weak regularity and consecutive topologizations and regularizations of pretopologies |
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Authors: | S Dolecki H-PA Künzi T Nogura |
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Institution: | a Institut de Mathématiques de Bourgogne, Université de Bourgogne, BP 47870, 21078 Dijon, France b Department of Mathematics, University of Cape Town, Rondebosch 7701, South Africa c Department of Mathematics, Ehime University, 790-Matsuyama, Ehime, Japan |
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Abstract: | L. Foged proved that a weakly regular topology on a countable set is regular. In terms of convergence theory, this means that the topological reflection Tξ of a regular pretopology ξ on a countable set is regular. It is proved that this still holds if ξ is a regular σ-compact pretopology. On the other hand, it is proved that for each n<ω there is a (regular) pretopology ρ (on a set of cardinality c) such that k(RT)ρ>n(RT)ρ for each k<n and n(RT)ρ is a Hausdorff compact topology, where R is the reflector to regular pretopologies. It is also shown that there exists a regular pretopology of Hausdorff RT-order ?ω0. Moreover, all these pretopologies have the property that all the points except one are topological and regular. |
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Keywords: | 54A05 54A20 54A25 54B30 54D10 |
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