Difference between revisions of "Manifold"
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| − | A manifold is a topological space locally homeomorphic to an open ball in some Euclidean space. The [[Whitney | + | A '''manifold''' is a [[topological space]] locally [[homeomorphic]] to an [[open set | open]] [[ball]] in some [[Euclidean space]]. Informally, this says that if one were living on a point in a manifold, the region surrounding any point would look just like "normal" Euclidean space, i.e. <math>\mathbb{R}^n</math> for some <math>n</math>. For example, the interior of a Mobius strip (that is, excluding its edge) or the surface of an infinite cylinder is a two-dimensional manifold because from each point on either surface the immediate neighborhood is indistinguishable from the usual [[Euclidean plane]], even though ''globally'' neither of these surfaces looks much like the plane. |
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| + | The [[Whitney Embedding Theorem]] allows us to visualise manifolds as being [[embedding | embedded]] in some Euclidean space. | ||
==Definition== | ==Definition== | ||
| − | A | + | A topological space <math>X</math> is said to be a manifold if and only if |
(i)<math>X</math> is [[Seperation axioms|Hausdorff]] | (i)<math>X</math> is [[Seperation axioms|Hausdorff]] | ||
| − | (ii)<math>X</math> is [[Countability|second-countable]], i.e. it has a [[countable]] [[base]]. | + | (ii)<math>X</math> is [[Countability|second-countable]], i.e. it has a [[countable]] [[base (topology) | base]]. |
{{stub}} | {{stub}} | ||
Revision as of 09:47, 6 April 2008
A manifold is a topological space locally homeomorphic to an open ball in some Euclidean space. Informally, this says that if one were living on a point in a manifold, the region surrounding any point would look just like "normal" Euclidean space, i.e. 
for some 
. For example, the interior of a Mobius strip (that is, excluding its edge) or the surface of an infinite cylinder is a two-dimensional manifold because from each point on either surface the immediate neighborhood is indistinguishable from the usual Euclidean plane, even though globally neither of these surfaces looks much like the plane.
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.
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