Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf sheaves simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

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.