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, or, equivalently, the closures of disjoint open subsets are disjoint.

Properties

The extremally disconnected spaces can be characterized among the Hausdorff spaces by the fact that any continuous function mapping a dense subspace of an extremally disconnected space to a compact Hausdorff space, can be extended to the whole space. They can also be characterized among the completely regular spaces by the fact that the lattice of continuous functions mapping an extremally dis connected space to the unit interval is complete (cf. (Giliman-Jerison, 1960), 3N and 6M). (Strauss 1967)

As projective objects

Theorem

(Gleason 1958)
Extremally disconnected topological spaces are precisely the projective objects in the category of compact Hausdorff topological spaces, and in fact in the category of regular Hausdorff topological spaces and proper maps between them.

See e.g. (Bhatt-Scholze 13, below theorem 1.8). For the second part see (Strauss 1967).

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).

In terms of logic

Remark

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 topological 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).

As a lifting property

Being extremally disconnected is equivalently a lifting property with respect to a surjective proper morphism of finite topological spaces (for more see the discussion at separation axioms in terms of lifting properties):

Namely that a topological space $X$ is extremally disconnected means equivalently that it solves the following lifting problems (from here):

Here boxes indicate open subsets in a finite topological space, and an arrow $x\to c$ means that $c$ is in the closure of $x$. See at Background and notation for more.

One may deduce Gleason’s theorem (Prop. ) in terms of these lifting properties:

Both being surjective and being proper are right lifting properties. Hence, the existence of a weak factorization system generated by this morphism implies that each space admits a surjective proper map from an extremally disconnected space.

For more see at separation axioms in terms of lifting properties the section extremally disconnected spaces being projective.

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

The result on projective spaces stems from

• Andrew M. Gleason, Projective topological spaces, Ill. J. Math. 2 no.4A (1958) pp.482-489. (euclid:ijm/1255454110)

• D. P. Strauss, (1967). Extremally Disconnected Spaces. Proceedings of the American Mathematical Society, 18(2), 305. doi:10.2307/2035286

Textbook account:

• Peter Johnstone, Stone Spaces

• L.Giliman, M.Jerison. Rings of continuous functions. Van Nostrand, 1960.

Discussion in the context of the pro-etale site:

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