# Difference between revisions of "Ultraconnected space"

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A [[topological space]] is termed an '''ultraconnected space''' if it satisfies the following equivalent conditions: | 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 is nonempty and cannot be expressed as a union of two proper open subsets |

− | # It cannot be expressed as a union of finitely many 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== |

## Revision as of 00:40, 5 January 2017

## Contents

## 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