Ultraconnected space: Difference between revisions

Revision as of 17:41, 28 January 2012

Definition

A topological space is termed an ultraconnected space if it is a non-empty space and an two non-empty disjoint closed subsets have non-empty intersection.

Relation with other properties

Weaker properties

Property	Proof of implication	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

T1 space: See ultraconnected and T1 implies one-point space

@@ Line 10: / Line 10: @@
 ! 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 space]] || || || ||
+| [[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}}
 |}