Connected manifold: Difference between revisions

Revision as of 19:23, 22 June 2016

This article describes a property of topological spaces obtained as a conjunction of the following two properties: connected space and manifold

This article defines a property of manifolds and hence also of topological spaces

Definition

A connected manifold is a topological space satisfying the following equiavlent conditions:

It is a connected space that is also a manifold.
It is a path-connected space that is also a manifold.

Relation with other properties

Stronger properties

Property	Meaning	Intermediate notions
compact connected manifold	connected and also a compact space	\|FULL LIST, MORE INFO
simply connected manifold	manifold that is also a simply connected space	\|FULL LIST, MORE INFO
compact connected orientable manifold		\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
homogeneous space	connected manifold implies homogeneous		\|FULL LIST, MORE INFO
manifold				\|FULL LIST, MORE INFO

@@ Line 1: / Line 1: @@
-{{topospace property}}
+{{topospace property conjunction|connected space|manifold}}
 {{manifold property}}
@@ Line 5: / Line 5: @@
 ==Definition==
-A '''connected manifold''' is a [[connected space]] which is also a manifold.
+A '''connected manifold''' is a topological space satisfying the following equiavlent conditions:
-Note that since any manifold is [[locally path-connected space|locally path-connected]], a connected manifold is also path-connected.
+# It is a [[connected space]] that is also a [[manifold]].
+# It is a [[path-connected space]] that is also a [[manifold]].
 ==Relation with other properties==
@@ Line 13: / Line 14: @@
 ===Stronger properties===
-* [[Compact connected manifold]]
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Weaker than::compact connected manifold]] || connected and also a [[compact space]] || || || {{intermediate notions short|connected manifold|compact connected manifold}}
+|-
+| [[Weaker than::simply connected manifold]] || manifold that is also a [[simply connected space]] || || || {{intermediate notions short|connected manifold|simply connected manifold}}
+|-
+| [[Weaker than::compact connected orientable manifold]] || || || || {{intermediate notions short|connected manifold|compact connected orientable manifold}}
+|}
 ===Weaker properties===
-* [[Homogeneous space]]: {{proofat|[[Connected manifold implies homogeneous]]}}
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Stronger than::homogeneous space]] || [[connected manifold implies homogeneous]] || || {{intermediate notions short|homogeneous space|connected manifold}}
+|-
+| [[Stronger than::manifold]] || || || || {{intermediate notions short|manifold|connected manifold}}
+|}
+See also [[Manifold#Weaker properties]]