# nLab paracompact locale

### 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.

Created on June 16, 2021 at 16:43:44. See the history of this page for a list of all contributions to it.