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 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 topologically the same as the usual [[Euclidean plane]], even though ''globally'' neither of these surfaces looks much like the plane.
  
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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The [[Whitney Embedding Theorem]] allows us to visualise manifolds as being [[embedding | embedded]] in some Euclidean space.  
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Note that the above describes a manifold in the topological category; in the smooth (analytic, holomorphic, etc) category, one would require the patching homeomorphisms to in fact be <math>C^{\infty}</math> (analytic, holomorphic, etc).
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There are also the generalizations of a manifold with boundary, a manifold with corners, and manifolds with even more funky singular points.
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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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*<math>X</math> is [[Separation axioms|Hausdorff]] (in general topology, this is not true, so non-Hausdorff manifolds indeed exist)
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*<math>X</math> is [[Countability|second-countable]], i.e. it has a [[countable]] [[base (topology) | base]].
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Latest revision as of 21:08, 13 October 2019

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. $\mathbb{R}^n$ for some $n$. 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 topologically the same as 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.

Note that the above describes a manifold in the topological category; in the smooth (analytic, holomorphic, etc) category, one would require the patching homeomorphisms to in fact be $C^{\infty}$ (analytic, holomorphic, etc).

There are also the generalizations of a manifold with boundary, a manifold with corners, and manifolds with even more funky singular points.

Definition

A topological space $X$ is said to be a manifold if and only if

  • $X$ is Hausdorff (in general topology, this is not true, so non-Hausdorff manifolds indeed exist)

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