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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