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.
Examples Two examples of hyperconnected spaces from point set topology are the cofinite topology on any infinite set and the right order topology on R {\displaystyle \mathbb {R} } .
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