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- Hyperconnected_space abstract "In mathematics, a hyperconnected space is a topological space X that cannot be written as the union of two proper closed sets. 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 disjointX cannot be written as the union of two proper closed setsevery nonempty open set is dense in Xthe interior of every proper closed set is emptyA space which satisfies any one of these conditions is called hyperconnected or irreducible. An irreducible set is a subset of a topological space for which the subspace topology is irreducible. Some authors do not consider the empty set to be irreducible (even though it vacuously satisfies the above conditions).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.Examples of hyperconnected spaces include the cofinite topology on any infinite space and the Zariski topology on an algebraic variety.Every hyperconnected space is both connected and locally connected (though not necessarily path-connected or locally path-connected). 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. A closed subspace need not be hyperconnected, however, the closure of any hyperconnected subspace is always hyperconnected.".
- Hyperconnected_space wikiPageID "7556651".
- Hyperconnected_space wikiPageRevisionID "604173911".
- Hyperconnected_space hasPhotoCollection Hyperconnected_space.
- Hyperconnected_space id "5813".
- Hyperconnected_space title "Hyperconnected space".
- Hyperconnected_space subject Category:Properties_of_topological_spaces.
- Hyperconnected_space type Abstraction100002137.
- Hyperconnected_space type Possession100032613.
- Hyperconnected_space type PropertiesOfTopologicalSpaces.
- Hyperconnected_space type Property113244109.
- Hyperconnected_space type Relation100031921.
- Hyperconnected_space comment "In mathematics, a hyperconnected space is a topological space X that cannot be written as the union of two proper closed sets.".
- Hyperconnected_space label "Espace topologique irréductible".
- Hyperconnected_space label "Espacio hiperconexo".
- Hyperconnected_space label "Espaço hiperconectado".
- Hyperconnected_space label "Hyperconnected space".
- Hyperconnected_space label "Irreduzibler topologischer Raum".
- Hyperconnected_space label "Przestrzeń nieprzywiedlna".
- Hyperconnected_space sameAs Irreduzibler_topologischer_Raum.
- Hyperconnected_space sameAs Espacio_hiperconexo.
- Hyperconnected_space sameAs Espace_topologique_irréductible.
- Hyperconnected_space sameAs Przestrzeń_nieprzywiedlna.
- Hyperconnected_space sameAs Espaço_hiperconectado.
- Hyperconnected_space sameAs m.02657y0.
- Hyperconnected_space sameAs Q1673182.
- Hyperconnected_space sameAs Q1673182.
- Hyperconnected_space sameAs Hyperconnected_space.
- Hyperconnected_space wasDerivedFrom Hyperconnected_space?oldid=604173911.
- Hyperconnected_space isPrimaryTopicOf Hyperconnected_space.