# Irreducible space

From Topospaces

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

## Contents

## Definition

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

### Weaker properties

### Incomparable properties

- Noetherian space:
*For full proof, refer: Irreducible not implies Noetherian*

### Opposite properties

## 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.