Irreducible space

This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

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.

Relation with other properties

Weaker properties

• Connected space

Incomparable properties

• Noetherian space: For full proof, refer: Irreducible not implies Noetherian

Opposite properties

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

Metaproperties

Hereditariness on open subsets

This property of topological spaces is hereditary on open subsets, or is open subspace-closed. In other words, any open subset of a topological space having this property, also has this property

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.