Manifold: Difference between revisions
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* It is [[Hausdorff space|Hausdorff]] | * It is [[Hausdorff space|Hausdorff]] | ||
* It is [[second countable space|second countable]] | * It is [[second-countable space|second-countable]] | ||
* | * It is [[locally Euclidean space|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 | ||
==Metaproperties== | ==Metaproperties== | ||
Revision as of 18:26, 26 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
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