In mathematics, an extremally disconnected space is a
topological space in which the closure of every open set is open. (The term "extremally disconnected" is correct, even though the word "extremally" does not appear in most dictionaries, and is sometimes mistaken by spellcheckers for the
homophone
A homophone () is a word that is pronounced the same (to varying extent) as another word but differs in meaning. A ''homophone'' may also differ in spelling. The two words may be spelled the same, for example ''rose'' (flower) and ''rose'' (p ...
''extremely disconnected''.)
An extremally disconnected space that is also
compact and
Hausdorff is sometimes called a Stonean space. This is not the same as a
Stone space, which is a
totally disconnected compact Hausdorff space. Every Stonean space is a Stone space, but not vice versa. In the duality between Stone spaces and
Boolean algebras, the Stonean spaces correspond to the
complete Boolean algebras.
An extremally disconnected
first-countable collectionwise Hausdorff space In mathematics, in the field of topology, a topological space X is said to be collectionwise Hausdorff if given any closed discrete subset of X, there is a pairwise disjoint family of open sets with each point of the discrete subset contained in exa ...
must be
discrete. In particular, for
metric spaces, the property of being extremally disconnected (the closure of every open set is open) is equivalent to the property of being discrete (every set is open).
Examples
* Every
discrete space
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
is extremally disconnected. Every
indiscrete space is both extremally disconnected and connected.
* The
Stone–Čech compactification of a discrete space is extremally disconnected.
* The
spectrum of an
abelian von Neumann algebra is extremally disconnected.
* Any commutative
AW*-algebra is isomorphic to
where
is extremally disconnected, compact and Hausdorff.
* Any infinite space with the
cofinite topology is both extremally disconnected and
connected. More generally, every
hyperconnected space is extremally disconnected.
* The space on three points with
base provides a
finite example of a space that is both extremally disconnected and connected. Another example is given by the
sierpinski space, since it is finite, connected, and hyperconnected.
Equivalent characterizations
A theorem due to says that the
projective objects of the
category of compact Hausdorff spaces are exactly the extremally disconnected compact Hausdorff spaces. A simplified proof of this fact is given by .
A compact Hausdorff space is extremally disconnected if and only if it is a
retract of the Stone–Čech compactification of a discrete space.
Applications
proves the
Riesz–Markov–Kakutani representation theorem
In mathematics, the Riesz–Markov–Kakutani representation theorem relates linear functionals on spaces of continuous functions on a locally compact space to measures in measure theory. The theorem is named for who introduced it for continuou ...
by reducing it to the case of extremally disconnected spaces, in which case the representation theorem can be proved by elementary means.
See also
*
Totally disconnected space
References
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* {{Citation, last=Semadeni, first=Zbigniew, title=Banach spaces of continuous functions. Vol. I, publisher=PWN---Polish Scientific Publishers, Warsaw, year=1971, mr=0296671
Properties of topological spaces