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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeDec 17th 2016
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI notice that the basic axioms for all three of the [[proximity space]] relations $\delta,\bowtie,\ll$ can be stated without any reference to points, only to the lattice structure of $P(X)$. But surprisingly, I am not aware of anyone having written down a notion of "proximity locale". Classically, are any of the relations $\delta,\bowtie,\ll$ determined by their restriction to opens (or closeds) in the underlying topology? If so, one could simply define a proximity locale by adding one of these relations to the frame of opens (or the coframe of closeds) of a locale and writing down the same axioms. If not, then perhaps it would work to use instead the lattice of sublocales.

   I notice that the basic axioms for all three of the proximity space relations $\delta,\bowtie,\ll$ can be stated without any reference to points, only to the lattice structure of $P(X)$. But surprisingly, I am not aware of anyone having written down a notion of “proximity locale”. Classically, are any of the relations $\delta,\bowtie,\ll$ determined by their restriction to opens (or closeds) in the underlying topology? If so, one could simply define a proximity locale by adding one of these relations to the frame of opens (or the coframe of closeds) of a locale and writing down the same axioms. If not, then perhaps it would work to use instead the lattice of sublocales.