nLab
Boolean locale
Context
Topos Theory
topos theory

Background
Toposes
Internal Logic
Topos morphisms
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

open subset , closed subset , neighbourhood

topological space , locale

base for the topology , neighbourhood base

finer/coarser topology

closure , interior , boundary

separation , sobriety

continuous function , homeomorphism

uniformly continuous function

embedding

open map , closed map

sequence , net , sub-net , filter

convergence

category Top

Universal constructions

Extra stuff, structure, properties

nice topological space

metric space , metric topology , metrisable space

Kolmogorov space , Hausdorff space , regular space , normal space

sober space

compact space , proper map

sequentially compact , countably compact , locally compact , sigma-compact , paracompact , countably paracompact , strongly compact

compactly generated space

second-countable space , first-countable space

contractible space , locally contractible space

connected space , locally connected space

simply-connected space , locally simply-connected space

cell complex , CW-complex

pointed space

topological vector space , Banach space , Hilbert space

topological group

topological vector bundle , topological K-theory

topological manifold

Examples

empty space , point space

discrete space , codiscrete space

Sierpinski space

order topology , specialization topology , Scott topology

Euclidean space

cylinder , cone

sphere , ball

circle , torus , annulus , Moebius strip

polytope , polyhedron

projective space (real , complex )

classifying space

configuration space

path , loop

mapping spaces : compact-open topology , topology of uniform convergence

Zariski topology

Cantor space , Mandelbrot space

Peano curve

line with two origins , long line , Sorgenfrey line

K-topology , Dowker space

Warsaw circle , Hawaiian earring space

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Boolean locale
Definition
A locale $L$ is Boolean if all of its opens (i.e. elements of the corresponding frame ) are complemented , i.e., for any $a\in O(L)$ we have $a\vee\neg a=1$ . Equivalently, we could say that for all $a\in A$ we have $a=\neg\neg a$ .

Properties
The underlying frames of Boolean locales are precisely complete Boolean algebras .

Maps of Boolean locales are automatically open . Their underlying morphisms of frames are precisely complete Boolean homomorphisms? , i.e., suprema-preserving homomorphisms of Boolean algebras .

Thus, the opposite category of Boolean locales is precisely the category of complete Boolean algebras and complete homomorphisms thereof.

By Stonean duality the category of Boolean locales is equivalent to the category of Stonean locales and open maps thereof. (Not every map of locales between Stonean locales is open, unlike for Boolean locales.)

Any locale $L$ has a maximal dense sublocale , whose opens are precisely the regular elements of $O(L)$ , i.e., elements $a\in O(L)$ such that $a=\neg\neg a$ . This sublocale is Boolean and is also known as the double negation sublocale .