Contents

# 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 $R$ 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.

###### Counter-Example

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

## 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 January 3, 2021 at 12:43:14. See the history of this page for a list of all contributions to it.