HOME

TheInfoList



OR:

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 \mathbb. 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
\text\left( \frac \right) , \text\left( \frac \right)
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 divisor
\text\left( \frac \right)
since the underlying space is the union of the affine planes \mathbb^2_, \mathbb^2_, and \mathbb^2_. Another non-example is given by the scheme
\text\left( \frac \right)
where f_4 is an irreducible degree 4 homogeneous polynomial. This is the union of the two genus 3 curves (by the genus–degree formula)
\text\left( \frac \right), \text \text\left( \frac \right)


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 U\subset X be an open subset. Any two disjoint open subsets of U would themselves be disjoint open subsets of X. So at least one of them must be empty.'' * More generally, every dense subset of a hyperconnected space is hyperconnected. : Proof: ''Suppose S is a dense subset of X and S=S_1\cup S_2 with S_1, S_2 closed in S. Then X=\overline S=\overline\cup\overline. Since X is hyperconnected, one of the two closures is the whole space X, say \overline=X. This implies that S_1 is dense in S, and since it is closed in S, it must be equal to S.'' *A closed subspace of a hyperconnected space need not be hyperconnected. : Counterexample: ''\Bbbk^2 with \Bbbk 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 V=Z(XY)=Z(X)\cup Z(Y)\subset\Bbbk^2 is closed and not hyperconnected.'' *The closure of any irreducible set is irreducible. : Proof: ''Suppose S\subseteq X where S is irreducible and write \operatorname_X(S)=F\cup G for two closed subsets F,G\subseteq \operatorname_X(S) (and thus in X). F':=F\cap S,\,G':=G\cap S are closed in S and S=F'\cup G' which implies S\subseteq F or S\subseteq G, but then \operatorname_X(S)=F or \operatorname_X(S)=G by definition of closure.'' *A space X which can be written as X=U_1\cup U_2 with U_1,U_2\subset X open and irreducible such that U_1\cap U_2\ne\emptyset is irreducible. : Proof: ''Firstly, we notice that if V is a non-empty open set in X then it intersects both U_1 and U_2; indeed, suppose V_1:=U_1\cap V\ne\emptyset, then V_1 is dense in U_1, thus \exists x\in\operatorname_(V_1)\cap U_2=U_1\cap U_2\ne\emptyset and x\in U_2 is a point of closure of V_1 which implies V_1\cap U_2\ne\emptyset and a fortiori V_2:=V\cap U_2\ne\emptyset. Now V=V\cap(U_1\cup U_2)=V_1\cup V_2 and taking the closure \operatorname_(V)\supseteq_(V_1)\cup_(V_2)=U_1\cup U_2=X, therefore V is a non-empty open and dense subset of X. Since this is true for every non-empty open subset, X 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