Irreducible space: Difference between revisions
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==Definition== | ==Definition== | ||
A [[topological space]] is said to be '''irreducible''' if it satisfies the following equivalent conditions: | 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 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 any two nonempty open subsets have nonempty intersection. | ||
# It is nonempty and every nonempty open subset is dense. | # It is nonempty and every nonempty open subset is dense. | ||
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===Weaker properties=== | ===Weaker properties=== | ||
* [[Connected space]] | * [[Connected space]] | ||
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* [[Noetherian space]]: {{proofat|[[Irreducible not implies Noetherian]]}} | * [[Noetherian space]]: {{proofat|[[Irreducible not implies Noetherian]]}} | ||
===Opposite properties=== | |||
* [[Hausdorff space]]: See [[irreducible and Hausdorff implies one-point space]] | |||
==Metaproperties== | ==Metaproperties== | ||
Latest revision as of 00:41, 5 January 2017
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:
- 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.