Manifold

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 $m$, then we call the manifold a $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$ is a fiber bundle with base space $B$ and fiber space $F$, and both $B$ and $F$ are manifolds, then $E$ is also a manifold. Note that this covers the particular cases of direct products and covering spaces.

References

• Topology (2nd edition) by James R. Munkres^{More info}, Page 225, Chapter 4, Section 36 (formal definition, as definition of $m$-manifold, where $m$ is the dimension)