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
A topological space is said to be irreducible or hyperconnected if it satisfies the following equivalent conditions:
- It is nonempty and cannot be expressed as a union of two proper closed subsets.
- It is nonempty and cannot be expressed as a union of finitely many proper closed subsets.
- It is nonempty and any two nonempty open subsets have nonempty intersection.
- It is nonempty and every nonempty open subset is dense.
Relation with other properties
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.