Adding the Hausdorff condition can make several properties become equivalent for a manifold. Indeed, a Hausdorff manifold is a locally compact Hausdorff space, hence it is completely regular.
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A manifold need not be connected, but every manifold M is a disjoint union of connected manifolds. These are just the connected components of M , which are open sets since manifolds are locally-connected.
Being locally path connected, a manifold is path-connected if and only if it is connected. It follows that the path-components are the same as the components. The Hausdorff property is not a local one; so even though Euclidean space is Hausdorff, a locally Euclidean space need not be. It is true, however, that every locally Euclidean space is T 1. An example of a non-Hausdorff locally Euclidean space is the line with two origins. This space is created by replacing the origin of the real line with two points, an open neighborhood of either of which includes all nonzero numbers in some open interval centered at zero.
This space is not Hausdorff because the two origins cannot be separated. A manifold is metrizable if and only if it is paracompact. Since metrizability is such a desirable property for a topological space, it is common to add paracompactness to the definition of a manifold.
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In any case, non-paracompact manifolds are generally regarded as pathological. An example of a non-paracompact manifold is given by the long line.
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Paracompact manifolds have all the topological properties of metric spaces. In particular, they are perfectly normal Hausdorff spaces. Manifolds are also commonly required to be second-countable. This is precisely the condition required to ensure that the manifold embeds in some finite-dimensional Euclidean space. Every second-countable manifold is paracompact, but not vice versa.
However, the converse is nearly true: a paracompact manifold is second-countable if and only if it has a countable number of connected components. In particular, a connected manifold is paracompact if and only if it is second-countable. Every second-countable manifold is separable and paracompact. Moreover, if a manifold is separable and paracompact then it is also second-countable. Every compact manifold is second-countable and paracompact.
Being an n -manifold is a topological property , meaning that any topological space homeomorphic to an n -manifold is also an n -manifold. By definition, every point of a locally Euclidean space has a neighborhood homeomorphic to an open subset of R n. Such neighborhoods are called Euclidean neighborhoods. It follows from invariance of domain that Euclidean neighborhoods are always open sets.
One can always find Euclidean neighborhoods that are homeomorphic to "nice" open sets in R n. Indeed, a space M is locally Euclidean if and only if either of the following equivalent conditions holds:. A Euclidean neighborhood homeomorphic to an open ball in R n is called a Euclidean ball.
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Euclidean balls form a basis for the topology of a locally Euclidean space. A space M is locally Euclidean if and only if it can be covered by Euclidean neighborhoods. A set of Euclidean neighborhoods that cover M , together with their coordinate charts, is called an atlas on M. The terminology comes from an analogy with cartography whereby a spherical globe can be described by an atlas of flat maps or charts.
Such a map is a homeomorphism between open subsets of R n.
That is, coordinate charts agree on overlaps up to homeomorphism. Different types of manifolds can be defined by placing restrictions on types of transition maps allowed. For example, for differentiable manifolds the transition maps are required to be diffeomorphisms. A 0-manifold is just a discrete space. A discrete space is second-countable if and only if it is countable. Every nonempty, paracompact, connected 1-manifold is homeomorphic either to R or the circle.
Every nonempty, compact, connected 2-manifold or surface is homeomorphic to the sphere , a connected sum of tori , or a connected sum of projective planes. A classification of 3-manifolds results from Thurston's geometrization conjecture , proven by Grigori Perelman in More specifically, Perelman's results provide an algorithm for deciding if two three-manifolds are homeomorphic to each other.
The full classification of n -manifolds for n greater than three is known to be impossible; it is at least as hard as the word problem in group theory , which is known to be algorithmically undecidable. In fact, there is no algorithm for deciding whether a given manifold is simply connected.
A slightly more general concept is sometimes useful. A period of rapid progress with TOP manifolds ensued, including, in , Siebenmann's refutation of the Hauptvermutung and the Triangulation Conjecture. Here is the first connected account of Kirby's and Siebenmann's basic research in this area. The five sections of this book are introduced by three articles by the authors that initially appeared between and Appendices provide a full discussion of the classification of homotopy tori, including Casson's unpublished work and a consideration of periodicity in topological surgery.
Robion C. Siebenmann , Lislot Siebenmann. Some basic theorems about topological manifolds by L Siebenmann. Stable classification of smooth and piecewise linear manifold struc.