In mathematics, an extremally disconnected space is a
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
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 as another word but differs in meaning or in spelling. The two words may be spelled the same, for example ''rose'' (flower) and ''rose'' (past tense of "rise"), or spelled differently, a ...
''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, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
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 variable (mathematics), variables are the truth values ''true'' and ''false'', usually denot ...
s, the Stonean spaces correspond to the
complete Boolean algebras.
An extremally disconnected
first-countable
In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base ...
collectionwise Hausdorff space 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, ...
. In particular, for
metric space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
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 and non-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 In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the conseque ...
is both extremally disconnected and connected.
* The
Stone–Čech compactification
In the mathematical discipline of general topology, Stone–Čech compactification (or Čech–Stone compactification) is a technique for constructing a Universal property, universal map from a topological space ''X'' to a Compact space, compact Ha ...
of a discrete space is extremally disconnected.
* The
spectrum
A spectrum (: spectra or spectrums) is a set of related ideas, objects, or properties whose features overlap such that they blend to form a continuum. The word ''spectrum'' was first used scientifically in optics to describe the rainbow of co ...
of an
abelian von Neumann algebra In functional analysis, a branch of mathematics, an abelian von Neumann algebra is a von Neumann algebra of operators on a Hilbert space in which all elements commutative, commute.
The prototypical example of an abelian von Neumann algebra is th ...
is extremally disconnected.
* Any commutative
AW*-algebra is isomorphic to
, for some space
which is extremally disconnected, compact and Hausdorff.
* Any infinite space with the
cofinite topology is both extremally disconnected and
connected. More generally, every
hyperconnected space
In the mathematical field of topology, a hyperconnected space or irreducible space is a topological space ''X'' that cannot be written as the union of two proper closed subsets (whether disjoint or non-disjoint). The name ''irreducible space'' is ...
is extremally disconnected.
* The space on three points with
base provides a
finite
Finite may refer to:
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked for person and/or tense or aspect
* "Finite", a song by Sara Gr ...
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.
The following spaces are not extremally disconnected:
* The
Cantor set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and mentioned by German mathematician Georg Cantor in 1883.
Throu ...
is not extremally disconnected. However, it is totally disconnected.
Equivalent characterizations
A theorem due to says that the
projective object In category theory, the notion of a projective object generalizes the notion of a projective module. Projective objects in abelian categories are used in homological algebra. The dual notion of a projective object is that of an injective object ...
s of the
category
Category, plural categories, may refer to:
General uses
*Classification, the general act of allocating things to classes/categories Philosophy
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce)
* 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 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
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) ...
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