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In the mathematical field of
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, a hyperconnected space or irreducible 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 po ...
''X'' that cannot be written as the union of two proper closed sets (whether disjoint or non-disjoint). The name ''irreducible space'' is preferred in algebraic geometry. For a topological space ''X'' the following conditions are equivalent: * No two nonempty
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
s are disjoint. * ''X'' cannot be written as the union of two proper closed sets. * Every nonempty open set is
dense Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
in ''X''. * The interior of every proper closed set 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. 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'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
is irreducible. Some authors do not consider the empty set to be irreducible (even though it vacuously satisfies the above conditions).


Examples

Two examples of hyperconnected spaces from
point set topology In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geomet ...
are the
cofinite topology In mathematics, a cofinite subset of a set X is a subset A whose complement in X is a finite set. In other words, A contains all but finitely many elements of X. If the complement is not finite, but it is countable, then one says the set is cocoun ...
on any infinite set and the
right order topology In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets. If ''X'' is a totally ordered set, th ...
on \mathbb. In algebraic geometry, taking the spectrum of a ring whose
reduced ring In ring theory, a branch of mathematics, a ring is called a reduced ring if it has no non-zero nilpotent elements. Equivalently, a ring is reduced if it has no non-zero elements with square zero, that is, ''x''2 = 0 implies ''x'' = ...
is an
integral domain In mathematics, specifically abstract algebra, 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 s ...
is an irreducible topological space—applying the
lattice theorem In group theory, the correspondence theorem (also the lattice theorem,W.R. Scott: ''Group Theory'', Prentice Hall, 1964, p. 27. and variously and ambiguously the third and fourth isomorphism theorem ) states that if N is a normal subgroup of ...
to the nilradical, which is within every prime, to show the spectrum of the quotient map is a homeomorphism, 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 In classical algebraic geometry, the genus–degree formula relates the degree ''d'' of an irreducible plane curve C with its arithmetic genus ''g'' via the formula: :g=\frac12 (d-1)(d-2). Here "plane curve" means that C is a closed curve in the pr ...
)
\text\left( \frac \right), \text \text\left( \frac \right)


Hyperconnectedness vs. connectedness

Every hyperconnected space is both
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
and
locally connected In topology and other branches of mathematics, a topological space ''X'' is locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets. Background Throughout the history of topology, connectedness ...
(though not necessarily path-connected 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 entirely of open, connected sets. Background Throughout the history of topology, connectedness ...
). 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 Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
and
locally connected In topology and other branches of mathematics, a topological space ''X'' is locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets. Background Throughout the history of topology, connectedness ...
(though not necessarily path-connected 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 entirely of open, connected sets. Background Throughout the history of topology, connectedness ...
). *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 In mathematics, in the field of topology, a topological space is said to be pseudocompact if its image under any continuous function to R is bounded. Many authors include the requirement that the space be completely regular in the definition of ps ...
. *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 (thus infinite) is hyperconnected in the
Zariski topology In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
, 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 algebraic geometry, an irreducible algebraic set or irreducible variety is an algebraic set that cannot be written as the union of two proper algebraic subsets. An irreducible component is an algebraic subset that is irreducible and maximal ( ...
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 Partition may refer to: Computing Hardware * Disk partitioning, the division of a hard disk drive * Memory partition, a subdivision of a computer's memory, usually for use by a single job Software * Partition (database), the division of a ...
). 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 ...
s. Since every irreducible space is connected, the irreducible components will always lie in the connected components. Every
Noetherian topological space In mathematics, a Noetherian topological space, named for Emmy Noether, is a topological space in which closed subsets satisfy the descending chain condition. Equivalently, we could say that the open subsets satisfy the ascending chain condition, ...
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 closed subset of ''X'' is the closure of exactly one point of ''X'': that is, every irreducible closed subset has a unique generic point. Definitio ...
* Geometrically irreducible


Notes


References

* *{{planetmath reference, urlname=HyperconnectedSpace, title=Hyperconnected space Properties of topological spaces