nLab
extremally disconnected topological space

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 $R$ a w-contractible ring, the profinite set $\pi_0(Spec R)$ 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 $Sh(X)$ of sheaves on a space $X$ is a De Morgan topos precisely when $X$ is extremally disconnected and this implies that all subobject lattices $sub(X)$ are De Morgan Heyting algebras, in particular $sub(1)$ corresponding to the frame of opens of $X$ (cf. De Morgan topos for details and references).

Examples

Example

For $S$ 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 disconnected.

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

Related concepts

• totally disconnected space

• De Morgan topos

References

• Peter Johnstone, Stone Spaces

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

• Bhargav Bhatt, Peter Scholze, The pro-étale topology for schemes (arXiv:1309.1198)

The result on projective spaces stems from

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