extremally disconnected topological space



topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory




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


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)


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).



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)


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)


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 March 16, 2018 at 08:43:51. See the history of this page for a list of all contributions to it.