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 preferred in algebraic geometry. For a topological space X the following conditions are equivalent: * No two nonempty open sets are disjoint. * X cannot be written as the union of two proper closed subsets. * 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 is vacuously 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, 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 is irreducible.

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 preferred in algebraic geometry. For a topological space X the following conditions are equivalent: No two nonempty open sets are disjoint. X cannot be written as the union of two proper closed subsets. 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 is vacuously 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, 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 is irreducible. 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{S_1}\cup\overline{S_2} . Since X is hyperconnected, one of the two closures is the whole space X , say \overline{S_1}=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, 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{Cl}_X(S)=F\cup G for two closed subsets F,G\subseteq \operatorname{Cl}_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{Cl}_X(S)=F or \operatorname{Cl}_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{Cl}_{U_1}(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{Cl}_{X}(V)\supseteq{\operatorname{Cl}}_{U_1}(V_1)\cup{\operatorname{Cl}}_{U_2}(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 sets. 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 Geometrically irreducible ==Notes== ==References==

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