Manifold: Difference between revisions
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* It is [[second countable space|second countable]] | * It is [[second countable space|second countable]] | ||
* 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 | * 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== | |||
{{DP-closed}} | |||
A direct product of manifolds is again a manifold. {{fillin}} | |||
Revision as of 14:33, 22 May 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
- 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 arbitrary products
View all properties of topological spaces closed under products
A direct product of manifolds is again a manifold. Fill this in later