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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeJan 9th 2016
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexAt [[complete space]] a construction of a completion of a space is sketched whose points are Cauchy filters, of which one can take the Hausdorff quotient to obtain the more usual sort of completion. This matches the procedure used in the common sequential completion of a metrirc space. However, if I understand it correctly, Vickers' [[localic completion]] of a metric space uses a different approach: rather than quotient the space of Cauchy filters, he restricts to the subspace of Cauchy filters that are also *rounded* (every open ball in the filter has a strict refinement in the filter, where $B(x,\delta)$ strictly refines $B(y,\epsilon)$ if $d(x,y)+\delta \lt \epsilon$). How generally does this work? At the moment I can't see how to make it work even for uniform spaces, let alone Cauchy spaces.

   At complete space a construction of a completion of a space is sketched whose points are Cauchy filters, of which one can take the Hausdorff quotient to obtain the more usual sort of completion. This matches the procedure used in the common sequential completion of a metrirc space. However, if I understand it correctly, Vickers’ localic completion of a metric space uses a different approach: rather than quotient the space of Cauchy filters, he restricts to the subspace of Cauchy filters that are also rounded (every open ball in the filter has a strict refinement in the filter, where $B(x,\delta)$ strictly refines $B(y,\epsilon)$ if $d(x,y)+\delta \lt \epsilon$). How generally does this work? At the moment I can’t see how to make it work even for uniform spaces, let alone Cauchy spaces.