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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| − | + | 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 | + | ==Definition== |
| + | A [[Topological space]] <math>X</math> is said to be a '''Manifold''' if and only if | ||
| + | |||
| + | (i)<math>X</math> is [[Seperation axioms|Hausdorff]] | ||
| + | |||
| + | (ii)<math>X</math> is [[Countability|second-countable]], i.e. it has a [[countable]] [[base]]. | ||
Revision as of 06:46, 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 
is said to be a Manifold if and only if
(i) is Hausdorff
(ii) is second-countable, i.e. it has a countable base.
