In the mathematical field of
topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
, a hyperconnected space or irreducible 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 ...
''X'' that cannot be written as the union of two proper closed subsets (whether disjoint or non-disjoint). The name ''irreducible space'' is preferred in
algebraic geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
.
For a topological space ''X'' the following conditions are equivalent:
* No two nonempty
open set
In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line.
In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...
s are
disjoint.
* ''X'' cannot be written as the union of two proper
closed subset
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a cl ...
s.
* Every nonempty open set is
dense in ''X''.
* Every open set is connected.
* The
interior of every proper closed subset of ''X'' is empty.
* Every subset is dense or
nowhere dense in ''X''.
* No two points can be separated by disjoint neighbourhoods.
A space which satisfies any one of these conditions is called ''hyperconnected'' or ''irreducible''. Due to the condition about neighborhoods of distinct points being in a sense the opposite of the
Hausdorff property, some authors call such spaces anti-Hausdorff.
The
empty set
In mathematics, the empty set or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exi ...
is
vacuously
In mathematics and logic, a vacuous truth is a conditional or universal statement (a universal statement that can be converted to a conditional statement) that is true because the antecedent cannot be satisfied.
It is sometimes said that a s ...
a hyperconnected or irreducible space under the definition above (because it contains no nonempty open sets). However some authors, especially those interested in applications to
algebraic geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
, add an explicit condition that an irreducible space must be nonempty.
An irreducible set is a subset of a topological space for which the
subspace topology
In topology and related areas of mathematics, a subspace of a topological space (''X'', ''𝜏'') is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''𝜏'' called the subspace topology (or the relative topology ...
is irreducible.
Examples
Two examples of hyperconnected spaces from
point set topology are the
cofinite topology on any
infinite set
In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable.
Properties
The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only set ...
and the
right order topology on
.
In algebraic geometry, taking the
spectrum of a ring whose
reduced ring is an
integral domain
In mathematics, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibilit ...
is an irreducible topological space—applying the
lattice theorem to the
nilradical, which is within every prime, to show the spectrum of the quotient map is a
homeomorphism
In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...
, this reduces to the irreducibility of the spectrum of an integral domain. For example, the
schemes
,
are irreducible since in both cases the polynomials defining the ideal are irreducible polynomials (meaning they have no non-trivial factorization). A non-example is given by the
normal crossing divisorsince the underlying space is the union of the affine planes
,
, and
. Another non-example is given by the scheme
where
is an irreducible degree 4
homogeneous polynomial. This is the union of the two genus 3 curves (by the
genus–degree formula)
Hyperconnectedness vs. connectedness
Every hyperconnected space is both
connected and
locally connected (though not necessarily
path-connected
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties t ...
or
locally path-connected
In topology and other branches of mathematics, a topological space ''X'' is
locally connected if every point admits a neighbourhood basis consisting of open connected sets.
As a stronger notion, the space ''X'' is locally path connected if e ...
).
Note that in the definition of hyper-connectedness, the closed sets don't have to be disjoint. This is in contrast to the definition of connectedness, in which the open sets are disjoint.
For example, the space of real numbers with the standard topology is connected but ''not'' hyperconnected. This is because it cannot be written as a union of two disjoint open sets, but it ''can'' be written as a union of two (non-disjoint) closed sets.
Properties
*The nonempty open subsets of a hyperconnected space are "large" in the sense that each one is dense in ''X'' and any pair of them intersects. Thus, a hyperconnected space cannot be
Hausdorff unless it contains only a single point.
*Every hyperconnected space is both
connected and
locally connected (though not necessarily
path-connected
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties t ...
or
locally path-connected
In topology and other branches of mathematics, a topological space ''X'' is
locally connected if every point admits a neighbourhood basis consisting of open connected sets.
As a stronger notion, the space ''X'' is locally path connected if e ...
).
*Since the closure of every non-empty open set in a hyperconnected space is the whole space, which is an open set, every hyperconnected space is
extremally disconnected.
*The
continuous image of a hyperconnected space is hyperconnected. In particular, any continuous function from a hyperconnected space to a Hausdorff space must be constant. It follows that every hyperconnected space is
pseudocompact.
*Every open subspace of a hyperconnected space is hyperconnected.
: Proof: ''Let
be an open subset. Any two disjoint open subsets of
would themselves be disjoint open subsets of
. So at least one of them must be empty.''
* More generally, every dense subset of a hyperconnected space is hyperconnected.
: Proof: ''Suppose
is a dense subset of
and
with
,
closed in
. Then
. Since
is hyperconnected, one of the two closures is the whole space
, say
. This implies that
is dense in
, and since it is closed in
, it must be equal to
.''
*A closed subspace of a hyperconnected space need not be hyperconnected.
: Counterexample: ''
with
an
algebraically closed field
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . In other words, a field is algebraically closed if the fundamental theorem of algebra ...
(thus infinite) is hyperconnected in the
Zariski topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology defined on geometric objects called varieties. It is very different from topologies that are commonly used in real or complex analysis; in particular, it is not ...
, while
is closed and not hyperconnected.''
*The
closure of any irreducible set is irreducible.
: Proof: ''Suppose
where
is irreducible and write
for two closed subsets
(and thus in
).
are closed in
and
which implies
or
, but then
or
by definition of
closure.''
*A space
which can be written as
with
open and irreducible such that
is irreducible.
: Proof: ''Firstly, we notice that if
is a non-empty open set in
then it intersects both
and
; indeed, suppose
, then
is dense in
, thus
and
is a
point of closure of
which implies
and a fortiori
. Now
and taking the closure
therefore
is a non-empty open and dense subset of
. Since this is true for every non-empty open subset,
is irreducible.''
Irreducible components
An
irreducible component in a topological space is a maximal irreducible subset (i.e. an irreducible set that is not contained in any larger irreducible set). The irreducible components are always closed.
Every irreducible subset of a space ''X'' is contained in a (not necessarily unique) irreducible component of ''X''. In particular, every point of ''X'' is contained in some irreducible component of ''X''. Unlike the
connected components of a space, the irreducible components need not be disjoint (i.e. they need not form a
partition). In general, the irreducible components will overlap.
The irreducible components of a Hausdorff space are just the
singleton set
In mathematics, a singleton (also known as a unit set or one-point set) is a set with exactly one element. For example, the set \ is a singleton whose single element is 0.
Properties
Within the framework of Zermelo–Fraenkel set theory, the a ...
s.
Since every irreducible space is connected, the irreducible components will always lie in the connected components.
Every
Noetherian topological space has finitely many irreducible components.
See also
*
Ultraconnected space
*
Sober space
In mathematics, a sober space is a topological space ''X'' such that every (nonempty) irreducible space, irreducible closed subset of ''X'' is the closure (topology), closure of exactly one point of ''X'': that is, every nonempty irreducible close ...
*
Geometrically irreducible
Notes
References
*
*
*{{planetmath reference, urlname=HyperconnectedSpace, title=Hyperconnected space
Properties of topological spaces