Hyperconnected space

In the mathematical field of topology, a hyperconnected space[1][2] or irreducible space[2] 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: 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.[3] The empty set is vacuously a hyperconnected or irreducible space under the definition above (because it contains no nonempty open sets).However some authors,[4] especially those interested in applications to algebraic geometry, add an explicit condition that an irreducible space must be nonempty.In algebraic geometry, taking the spectrum of a ring whose reduced ring is an integral domain 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, this reduces to the irreducibility of the spectrum of an integral domain.are irreducible since in both cases the polynomials defining the ideal are irreducible polynomials (meaning they have no non-trivial factorization).since the underlying space is the union of the affine planesNote 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.The irreducible components of a Hausdorff space are just the singleton sets.Every Noetherian topological space has finitely many irreducible components.
Hyperconnectivitytopologytopological spacealgebraic geometryopen setsdisjointclosed subsetsinteriornowhere denseHausdorffempty setvacuouslysubspace topologypoint set topologycofinite topologyspectrum of a ringreduced ringintegral domainlattice theoremnilradicalschemesnormal crossing divisorgenus–degree formulaconnectedlocally connectedpath-connectedlocally path-connectedextremally disconnectedcontinuouspseudocompactalgebraically closed fieldZariski topologyclosurepoint of closureirreducible componentconnected componentspartitionsingleton setsNoetherian topological spaceUltraconnected spaceSober spaceGeometrically irreducibleSteen, Lynn ArthurSeebach, J. Arthur Jr.Counterexamples in TopologySpringer-VerlagPlanetMath