For the computer networking term, see Hyperconnectivity. For hyper-connectivity in node-link graphs, see Connectivity (graph theory) § Super- and hyper-connectivity.
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.
^Steen & Seebach, p. 29
^ abHart, Nagata & Vaughan 2004, p. 9.
^Van Douwen, Eric K. (1993). "An anti-Hausdorff Fréchet space in which convergent sequences have unique limits". Topology and Its Applications. 51 (2): 147–158. doi:10.1016/0166-8641(93)90147-6.
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