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| * It is [[Hausdorff space|Hausdorff]] | | * It is [[Hausdorff space|Hausdorff]] |
| * It is [[second-countable space|second-countable]] | | * It is [[second-countable space|second-countable]] |
| * It is [[locally Euclidean space|locally Euclidean]], viz 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 (we usually assume that the same Euclidean space is used for all points, viz that the dimension is the same at all points) | | * It is [[locally Euclidean space|locally Euclidean]], viz 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 (we usually assume that the same Euclidean space is used for all points, viz., that the dimension is the same at all points) |
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| | If the dimension of the Euclidean space at each point is <math>m</math>, then we call the manifold a <math>m<math>-manifold. |
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| ==Significance of three parts of the definition== | | ==Significance of three parts of the definition== |
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| If <math>E</math> is a [[fiber bundle]] with base space <math>B</math> and fiber space <math>F</math>, and both <math>B</math> and <math>F</math> are manifolds, then <math>E</math> is also a manifold. Note that this covers the particular cases of direct products and covering spaces. | | If <math>E</math> is a [[fiber bundle]] with base space <math>B</math> and fiber space <math>F</math>, and both <math>B</math> and <math>F</math> are manifolds, then <math>E</math> is also a manifold. Note that this covers the particular cases of direct products and covering spaces. |
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| | ==References== |
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| | * {{booklink-defined|Munkres}}, Page 225, Chapter 4, Section 36 (formal definition, as definition of <math>m</math>-manifold, where <math>m</math> is the dimension) |
Revision as of 17:25, 20 July 2008
This article defines a property of topological space that is pivotal (viz important) among currently studied properties of topological spaces
The article on this topic in the Differential Geometry Wiki can be found at: topological manifold
Definition
A topological space is said to be a manifold if it satisfies the following equivalent conditions:
- It is Hausdorff
- It is second-countable
- It is locally Euclidean, viz 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 (we usually assume that the same Euclidean space is used for all points, viz., that the dimension is the same at all points)
If the dimension of the Euclidean space at each point is 
, then we call the manifold a ![{\displaystyle m<math>-manifold.==Significanceofthreepartsofthedefinition=====SignificanceoflocalEuclideanness===LocallyEuclideanisthemostimportantpropertyofmanifolds,sincethismeansthatallthenicepropertiesthatweknowaboutEuclideanspaces,areapplicablelocally.Thus,manifoldsarelocallycontractible,locallypath-connected,locallymetrizable,andsoon.Also,manypropertiesofthemanifoldthatareessentiallylocalinnaturecanbeprovedusinglocalEuclideanness.===SignificanceofHausdorffness===IfwedonotassumeHausdorffness,wegetpathologieslikethe[[linewithtwoorigins]].Morepertinently,theimportantwayinwhichweuseHausdorffnessisasfollows:inaHausdorffspace,anycompactsubsetisclosed.Thus,inparticular,theimagesofcloseddiscsofEuclideanspace,insidethemanifold,continuetoremainclosedinthewholemanifold.Thisiscrucialtoapplyingthe[[gluinglemmaforclosedsubsets]],forproofslikethoseofthefactthat[[connectedmanifoldimplieshomogeneous|anyconnectedmanifoldishomogeneous]]orthat[[manifoldimpliesnondegenerate|theinclusionofanypointinamanifoldisacofibration]].===Significanceofsecond-countability===Theassumptionofsecond-countabilitycanbedispensedwithforanumberofpurposes,butiscrucialforsomeapplications.Thestandardexampleofsomethingthatisamanifoldbutforthesecond-countabilityassumption,isthe[[longline]].==Relationwithotherproperties=====Strongerproperties===*[[Connectedmanifold]]*[[Compactmanifold]]*[[Differentiablemanifold]]*[[Liegroupasatopologicalspace|Liegroup]]===Weakerproperties===*[[Manifoldwithboundary]]*[[Closedsub-Euclideanspace]]:{proofat|[[Manifoldimpliesclosedsub-Euclidean]]}*[[Metrizablespace]]*[[ParacompactHausdorffspace]]*[[Normalspace]]*[[LocallyEuclideanspace]]*[[Locallycontractiblespace]]*[[Locallymetrizablespace]]*[[Nondegeneratespace]]:{proofat|[[Manifoldimpliesnondegenerate]]}*[[Compactlynondegeneratespace]]*[[Spacewithhomologyoffinitetype]]==Metaproperties=={finite-DP-closed}Adirectproductofmanifoldsisagainamanifold.{fillin}{coveringspace-closed}Any[[coveringspace]]ofamanifoldnaturallygetsthestructureofamanifold.{fiber-bundle-closed}If<math>E}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd939cf8ed8399d1bc28d531a2d933df54e0f0d7)
is a fiber bundle with base space 
and fiber space 
, and both and are manifolds, then 
is also a manifold. Note that this covers the particular cases of direct products and covering spaces.
References
- Topology (2nd edition) by James R. MunkresMore info, Page 225, Chapter 4, Section 36 (formal definition, as definition of -manifold, where is the dimension)