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

Symbol-free definition

A topological space is said to be irreducible if it is nonempty and cannot be expressed as a union of two proper closed subsets.

Relation with other properties

Weaker properties

• Supercompact space
• Compact space
• Connected space

Incomparable properties

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

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.