Ultraconnected space: Difference between revisions
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* [[T1 space]]: See [[ultraconnected and T1 implies one-point space]] | * [[T1 space]]: See [[ultraconnected and T1 implies one-point space]] | ||
===Similar properties=== | ===Similar properties=== | ||
* [[Irreducible space]], with a similar definition but the roles of "open" and "closed" interchanged | * [[Irreducible space]], with a similar definition but the roles of "open" and "closed" interchanged | ||
Latest revision as of 00:45, 5 January 2017
Definition
A topological space is termed an ultraconnected space if it satisfies the following equivalent conditions:
- It is nonempty and cannot be expressed as a union of two proper open subsets
- It is nonempty and cannot be expressed as a union of finitely many proper open subsets
- It is nonempty and any two nonempty closed subsets have nonempty intersection
Relation with other properties
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| path-connected space | ultraconnected implies path-connected | |FULL LIST, MORE INFO | ||
| connected space | |FULL LIST, MORE INFO | |||
| normal space | ultraconnected implies normal | |FULL LIST, MORE INFO | ||
| pseudocompact space | |FULL LIST, MORE INFO | |||
| limit point-compact space | |FULL LIST, MORE INFO |
Opposite properties
Similar properties
- Irreducible space, with a similar definition but the roles of "open" and "closed" interchanged