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 internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology 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 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.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJun 16th 2021
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexCreated with the following content: *** ### Definition A [[locale]] is __paracompact__ if it is [[regular]] and every open cover has a [[locally finite]] refinement. ### Properties Paracompact locales are very closely related to [[fully normal locales]]. In fact, for [[regular locales]] these two properties are equivalent. Any [[metrizable locale]] is paracompact. Any [[Lindelöf locale]] is paracompact. A [[locale]] is paracompact if and only if it admits a [[complete uniformity]]. The [[full subcategory]] of paracompact locales is a [[reflective subcategory]] of the category of [[completely regular locales]] as well as the category of all [[locales]. In particular, the inclusion functor from paracompact locales to locales preserves [[small limits]], so in particular, products of paracompact locales are paracompact. This last property clearly distinguishes paracompact locales from paracompact spaces, since products of paracompact spaces need not be paracompact. ### Related concets * [[fully normal locale]] * [[paracompact space]] * [[fully normal space]] *** <a href="https://ncatlab.org/nlab/revision/paracompact+locale/1">v1</a>, <a href="https://ncatlab.org/nlab/show/paracompact+locale">current</a>

   Created with the following content:

   ---

   Definition

   A locale is paracompact if it is regular and every open cover has a locally finite refinement.

   Properties

   Paracompact locales are very closely related to fully normal locales. In fact, for regular locales these two properties are equivalent.

   Any metrizable locale is paracompact.

   Any Lindelöf locale is paracompact.

   A locale is paracompact if and only if it admits a complete uniformity.

   The full subcategory of paracompact locales is a reflective subcategory of the category of completely regular locales as well as the category of all [[locales].

   In particular, the inclusion functor from paracompact locales to locales preserves small limits, so in particular, products of paracompact locales are paracompact.

   This last property clearly distinguishes paracompact locales from paracompact spaces, since products of paracompact spaces need not be paracompact.

   Related concets

   • fully normal locale

   • paracompact space

   • fully normal space

   ---

   v1, current