Hyperconnected space
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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:
- 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.
- 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.[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.
An irreducible set is a subset of a topological space for which the subspace topology is irreducible.