Difference between revisions of "Manifold"

(New page: Manifold A manifold is a topological space locally homeomorphic to an open ball in some Euclidean space. It has some other properties, like having a countable basis or something, but nobo...)
 
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Manifold
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A manifold is a topological space locally homeomorphic to an open ball in some Euclidean space. The [[Whitney embedding theorem]] allows us to visualise manifolds as being 'embedded' in some [[Euclidean space]].
  
A manifold is a topological space locally homeomorphic to an open ball in some Euclidean space. It has some other properties, like having a countable basis or something, but nobody really cares about these. You can go ahead and think about manifolds as subspaces of some large Euclidean space anyway, since we can do this by the Whitney embedding theorem.
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==Definition==
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A [[Topological space]] <math>X</math> is said to be a '''Manifold''' if and only if
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(i)<math>X</math> is [[Seperation axioms|Hausdorff]]
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(ii)<math>X</math> is [[Countability|second-countable]], i.e. it has a [[countable]] [[base]].

Revision as of 07:46, 6 April 2008

A manifold is a topological space locally homeomorphic to an open ball in some Euclidean space. The Whitney embedding theorem allows us to visualise manifolds as being 'embedded' in some Euclidean space.

Definition

A Topological space $X$ is said to be a Manifold if and only if

(i)$X$ is Hausdorff

(ii)$X$ is second-countable, i.e. it has a countable base.

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