Ultraconnected space: Difference between revisions
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==Definition== | ==Definition== | ||
A [[topological space]] is termed an '''ultraconnected space''' if it is a | 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== | ==Relation with other properties== | ||
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ||
|- | |- | ||
| [[Stronger than::path-connected space]] || || || || | | [[Stronger than::path-connected space]] || || [[ultraconnected implies path-connected]] || || {{intermediate notions short|path-connected space|ultraconnected space}} | ||
|- | |||
| [[Stronger than::connected space]] || || || || {{intermediate notions short|connected space|ultraconnected space}} | |||
|- | |- | ||
| [[Stronger than::normal | | [[Stronger than::normal space]] || || [[ultraconnected implies normal]] || || {{intermediate notions short|normal space|ultraconnected space}} | ||
|- | |- | ||
| [[Stronger than::pseudocompact space]] || || || || | | [[Stronger than::pseudocompact space]] || || || || {{intermediate notions short|pseudocompact space|ultraconnected space}} | ||
|- | |- | ||
| [[Stronger than::limit point-compact space]] || || || || | | [[Stronger than::limit point-compact space]] || || || || {{intermediate notions short|limit point-compact space|ultraconnected space}} | ||
|} | |} | ||
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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=== | |||
* [[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