Manifold: Difference between revisions
No edit summary |
|||
| Line 18: | Line 18: | ||
* [[Polyhedron]] | * [[Polyhedron]] | ||
* [[CW-space]] | * [[CW-space]] | ||
* [[Perfectly normal space]] (and all separation axioms sitting below it) | * [[Perfectly normal space]] (and all separation axioms sitting below it) | ||
* [[Locally contractible space]] | * [[Locally contractible space]] | ||
Revision as of 00:33, 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
- Locally Euclidean space
- Polyhedron
- CW-space
- Perfectly normal space (and all separation axioms sitting below it)
- Locally contractible space
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