Ultraconnected space: Difference between revisions
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==Definition== | ==Definition== | ||
A [[topological space]] is termed an '''ultraconnected space''' if it | A [[topological space]] is termed an '''ultraconnected space''' if it satisfies the following equivalent conditions: | ||
# It cannot be expressed as a union of two proper open subsets | |||
# It cannot be expressed as a union of finitely many proper open subsets | |||
# Any two nonempty closed subsets have nonempty intersection | |||
==Relation with other properties== | ==Relation with other properties== | ||
Revision as of 00:39, 5 January 2017
Definition
A topological space is termed an ultraconnected space if it satisfies the following equivalent conditions:
- It cannot be expressed as a union of two proper open subsets
- It cannot be expressed as a union of finitely many proper open subsets
- 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 |