Connected manifold: Difference between revisions
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{{topospace property}} | {{topospace property conjunction|connected space|manifold}} | ||
{{manifold property}} | {{manifold property}} | ||
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==Definition== | ==Definition== | ||
A '''connected manifold''' is a | 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== | ==Relation with other properties== | ||
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===Stronger properties=== | ===Stronger properties=== | ||
{| 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=== | ===Weaker properties=== | ||
{| 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}} | |||
|- | |||
| [[Stronger than::homology manifold]] || [[locally compact space]] whose homology groups with respect to the exclusion of any point look like those of a manifold || || || {{intermediate notions short|homology manifold|connected manifold}} | |||
|- | |||
| [[Stronger than::manifold with boundary]] || Hausdorff, second-countable, and every point is contained in an open subset that is homeomorphic to an open subset of Euclidean half-space || || | {{intermediate notions short|manifold with boundary|connected manifold}} | |||
|- | |||
| [[Stronger than::closed sub-Euclidean space]] || homeomorphic to a closed subset of Euclidean space || [[manifold implies closed sub-Euclidean]] || obvious counterexamples, such as a closed unit disk || {{intermediate notions short|closed sub-Euclidean space|connected manifold}} | |||
|- | |||
| [[Stronger than::metrizable space]] || underlying topological space of a [[metric space]] || || || {{intermediate notions short|metrizable space|connected manifold}} | |||
|- | |||
|[[Stronger than::paracompact Hausdorff space]] || [[paracompact space|paracompact]] and [[Hausdorff space|Hausdorff]] || (via metrizable) || || {{intermediate notions short|paracompact Hausdorff space|connected manifold}} | |||
|- | |||
| [[Stronger than::normal space]] || any two disjoint closed subsets can be separated by disjoint open subsets || || || {{intermediate notions short|normal space|connected manifold}} | |||
|- | |||
| [[Stronger than::regular space]] || any point and closed subset not containing it can be separated by disjoint open subsets || || || {{intermediate notions short|regular space|connected manifold}} | |||
|- | |||
| [[Stronger than::Hausdorff space]] || any two distinct points can be separated by disjoint open subsets || || || {{intermediate notions short|Hausdorff space|connected manifold}} | |||
|- | |||
| [[Stronger than::locally Euclidean space]] || every point is contained in an open subset that is homeomorphic to an open subset of Euclidean space || || || {{intermediate notions short|locally Euclidean space|connected manifold}} | |||
|- | |||
| [[Stronger than::locally contractible space]] || it has a basis of open subsets that are all contractible || || || {{intermediate notions short|locally contractible space|connected manifold}} | |||
|- | |||
| [[Stronger than::locally metrizable space]] || it has a basis of open subsets that are all metrizable || || || {{intermediate notions short|locally metrizable space|connected manifold}} | |||
|- | |||
| [[Stronger than::locally compact space]] || every point is contained in an open subset whose closure is compact || || || {{intermediate notions short|locally compact space|connected manifold}} | |||
|- | |||
| [[Stronger than::nondegenerate space]] || the inclusion of any point in it is a [[cofibration]] || [[manifold implies nondegenerate]] || || {{intermediate notions short|nondegenerate space|connected manifold}} | |||
|- | |||
|[[Stronger than::compactly nondegenerate space]] || every point is contained in an open subset whose closure is compact, and the inclusion of the point in the closure is a cofibration. || || || {{intermediate notions short|compactly nondegenerate space|connected manifold}} | |||
|} | |||
See also [[Manifold#Weaker properties]] | |||
Latest revision as of 19:27, 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 | Proof of implication | Proof of strictness (reverse implication failure) | 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 | |||
| homology manifold | locally compact space whose homology groups with respect to the exclusion of any point look like those of a manifold | Manifold|FULL LIST, MORE INFO | ||
| manifold with boundary | Hausdorff, second-countable, and every point is contained in an open subset that is homeomorphic to an open subset of Euclidean half-space | Manifold|FULL LIST, MORE INFO | ||
| closed sub-Euclidean space | homeomorphic to a closed subset of Euclidean space | manifold implies closed sub-Euclidean | obvious counterexamples, such as a closed unit disk | |FULL LIST, MORE INFO |
| metrizable space | underlying topological space of a metric space | Manifold|FULL LIST, MORE INFO | ||
| paracompact Hausdorff space | paracompact and Hausdorff | (via metrizable) | Manifold|FULL LIST, MORE INFO | |
| normal space | any two disjoint closed subsets can be separated by disjoint open subsets | Manifold|FULL LIST, MORE INFO | ||
| regular space | any point and closed subset not containing it can be separated by disjoint open subsets | Manifold|FULL LIST, MORE INFO | ||
| Hausdorff space | any two distinct points can be separated by disjoint open subsets | Manifold|FULL LIST, MORE INFO | ||
| locally Euclidean space | every point is contained in an open subset that is homeomorphic to an open subset of Euclidean space | Manifold|FULL LIST, MORE INFO | ||
| locally contractible space | it has a basis of open subsets that are all contractible | Manifold|FULL LIST, MORE INFO | ||
| locally metrizable space | it has a basis of open subsets that are all metrizable | Manifold|FULL LIST, MORE INFO | ||
| locally compact space | every point is contained in an open subset whose closure is compact | Manifold|FULL LIST, MORE INFO | ||
| nondegenerate space | the inclusion of any point in it is a cofibration | manifold implies nondegenerate | Manifold|FULL LIST, MORE INFO | |
| compactly nondegenerate space | every point is contained in an open subset whose closure is compact, and the inclusion of the point in the closure is a cofibration. | Manifold|FULL LIST, MORE INFO |
See also Manifold#Weaker properties