Irreducible space

Definition
A topological space is said to be irreducible or hyperconnected if it satisfies the following equivalent conditions:


 * 1) It is nonempty and cannot be expressed as a union of two proper closed subsets.
 * 2) It is nonempty and cannot be expressed as a union of finitely many proper closed subsets.
 * 3) It is nonempty and any two nonempty open subsets have nonempty intersection.
 * 4) It is nonempty and every nonempty open subset is dense.

Weaker properties

 * Connected space

Incomparable properties

 * Noetherian space:

Opposite properties

 * Hausdorff space: See irreducible and Hausdorff implies one-point space

Metaproperties
Any nonempty open subset of an irreducible space is irreducible.

If a dense subset of a topological space is irreducible, so is the whole space.