Hyperconnected space

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.

  1. ^ Steen & Seebach, p. 29
  2. ^ a b Hart, Nagata & Vaughan 2004, p. 9.
  3. ^ 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.
  4. ^ "Section 5.8 (004U): Irreducible components—The Stacks project".