nLab
spatial locale

Spatial locales

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Spatial locales

Idea

A spatial locale is a locale that comes from a topological space. This is an extra property of locales, a property of having enough points.

Definitions

Let XX be a topological space. Then we may define a locale, denoted Ω(X)\Omega(X), whose frame of opens is precisely the frame of open subspaces of XX.

A locale is spatial if it is isomorphic to Ω(X)\Omega(X) for some topological space XX.

A locale LL has enough points if, given any two opens UU and VV in LL, U=VU = V if (hence iff) precisely the same points of LL belong to UU as belong to VV.

Properties

The following conditions are all logically equivalent on a locale LL:

  1. LL is spatial, as defined above.
  2. LL has enough points, as defined above.
  3. Given any two opens UU and VV in LL, UVU \leq V if (hence iff) every point of LL that belongs to UU also belongs to VV.
  4. LL is isomorphic to Ω(pt(L))\Omega(pt(L)), where pt(L)pt(L) is the space of points? of LL.
  5. The natural morphism η L:Ω(pt(L))L\eta_L\colon \Omega(pt(L)) \to L (the counit of the adjunction from Top and Loc) is an isomorphism.

(It would be nice to state this as a theorem and put in a proof.)

Basically, what is going on here is that we have an idempotent adjunction from topological spaces to locales, and the spatial locales comprise the image of this adjunction. The corresponding condition on topological spaces is being sober.

Therefore, the full subcategory of LocLoc on the spatial locales is equivalent to the full subcategory of TopTop on sober spaces.

Terminology

The term ‘spatial locale’ can be confusing; it suggests a locale in Top or in some category Sp of spaces, which is not correct. Instead, the adjective ‘spatial’ should be taken in the same vein as ‘localic’ in ‘localic topos’ or ‘topological’ in ‘topological convergence’. These two terms also suggest that these other locales are not spaces, which is incorrect.

The really clear term for a spatial locale is ‘locale with enough points to separate the opens’, but ‘locale with enough points’ should be unambiguous. However, it is still a bit long. Occasionally one sees ‘spacial’ instead of ‘spatial’.

Criteria for spatiality

  1. Assuming the axiom of choice, locally compact locales are spatial. In particular, compact regular locales are locally compact, hence automatically spatial.

  2. Any coherent locale is also spatial.

  3. More generally, the meet of a countable family of open sublocales (i.e., a G δG_\delta-sublocale) of a compact regular locale is spatial.

  4. The completion of a uniform locale with a countable basis of uniformity is spatial.

  5. Stonean locales are spatial.

Last revised on February 3, 2021 at 14:26:49. See the history of this page for a list of all contributions to it.