In mathematics, an extremally disconnected space is a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
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
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
and
Hausdorff is sometimes called a Stonean space. This is not the same as a
Stone space, which is a
totally disconnected
In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty set) ...
compact Hausdorff space. Every Stonean space is a Stone space, but not vice versa. In the duality between Stone spaces and
Boolean algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in e ...
s, the Stonean spaces correspond to the
complete Boolean algebra
In mathematics, a complete Boolean algebra is a Boolean algebra in which every subset has a supremum (least upper bound). Complete Boolean algebras are used to construct Boolean-valued models of set theory in the theory of forcing. Every Boolea ...
s.
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
Discrete may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
* Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit
*Discrete group, a ...
. In particular, for
metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
s, 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
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors i ...
of an
abelian von Neumann algebra In functional analysis, an abelian von Neumann algebra is a von Neumann algebra of operators on a Hilbert space in which all elements commute.
The prototypical example of an abelian von Neumann algebra is the algebra ''L''∞(''X'', μ) for μ ...
is extremally disconnected.
* Any commutative
AW*-algebra In mathematics, an AW*-algebra is an algebraic generalization of a W*-algebra. They were introduced by Irving Kaplansky in 1951. As operator algebras, von Neumann algebras, among all C*-algebras, are typically handled using one of two means: they ...
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 continuo ...
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
*
*
*
*
*
* {{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