Manifold: Difference between revisions
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===Stronger properties=== | ===Stronger properties=== | ||
* [[Compact manifold]] | * [[Compact manifold]] | ||
* [[Paracompact manifold]] | * [[Paracompact manifold]] | ||
* [[Metrizable manifold]] | |||
* [[Differentiable manifold]] | |||
* [[Lie group as a topological space|Lie group]] | * [[Lie group as a topological space|Lie group]] | ||
Revision as of 01:12, 27 October 2007
This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces
Definition
A topological space is said to be a manifold if it satisfies the following equivalent conditions:
- It is Hausdorff
- It is second-countable
- It is locally Euclidean, viz every point has a neighbourhood that is homeomorphic to some open set in Euclidean space. If the topological space is connected, the dimension of the Euclidean space is the same for all points
Relation with other properties
Stronger properties
Weaker properties
- Manifold with boundary
- Locally Euclidean space
- Polyhedron
- CW-space
- Locally contractible space
- Space with finitely generated homology
Metaproperties
Products
This property of topological spaces is closed under taking finite products
A direct product of manifolds is again a manifold. Fill this in later