Difference between revisions of "Ultraconnected space"

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(Relation with other properties)
(Definition)
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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
# Any two nonempty closed subsets have nonempty intersection
+
# 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

Definition

A topological space is termed an ultraconnected space if it satisfies the following equivalent conditions:

  1. It is nonempty and cannot be expressed as a union of two proper open subsets
  2. It is nonempty and cannot be expressed as a union of finitely many proper open subsets
  3. 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