nLab
extremally disconnected topological space

Contents

Context

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

Contents

Definition

Definition

A topological space is called extremally disconnected if the closure of any open subset is still an open subset.

Properties

Theorem (Gleason)

Extremally disconnected topological spaces are precisely the projective objects in the category of compact Hausdorff topological spaces.

See e.g. (Bhatt-Scholze 13, below theorem 1.8)

Proposition

For RR a w-contractible ring, the profinite set π 0(SpecR)\pi_0(Spec R) is an extremally disconnected profinite set.

Part of (Bhatt-Scholze 13, theorem 1.8).

The logical side

The extremal disconnectedness of a space is correlated with the property that the frame of its open subsets is a De Morgan Heyting algebra hence with the validity of the De Morgan law in logic, since a result by Johnstone says that a topos Sh(X)Sh(X) of sheaves on a space XX is a De Morgan topos precisely when XX is extremally disconnected and this implies that all subobject lattices sub(X)sub(X) are De Morgan Heyting algebras, in particular sub(1)sub(1) corresponding to the frame of opens of XX (cf. De Morgan topos for details and references).

Examples

Example

For SS any set regarded as a discrete topological space, its Stone-Cech compactification is extremally disconnected.

(e.g. Bhatt-Scholze 13, example 2.4.6)

Counter-Example

The profinite set underlying the p-adic integers, regarded as a Stone space, is not extremally diconnected.

(e.g. Bhatt-Scholze 13, example 2.4.7)

References

Discussion in the context of the pro-etale site is in

The result on projective spaces stems from

  • Andrew M. Gleason, Projective topological spaces , Ill. J. Math. 2 no.4A (1958) pp.482-489. (euclid)

Last revised on April 29, 2019 at 13:51:40. See the history of this page for a list of all contributions to it.