Topological manifold
A topological space X is called locally Euclidean if there is a non-negative integer n such that every point in X has a neighborhood which is homeomorphic to real n-space Rn.
[2] A topological manifold is a locally Euclidean Hausdorff space.
An n-manifold will mean a topological manifold such that every point has a neighborhood homeomorphic to Rn.
Manifolds inherit many of the local properties of Euclidean space.
Adding the Hausdorff condition can make several properties become equivalent for a manifold.
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.
The long line is an example a normal Hausdorff 1-dimensional topological manifold that is not metrizable nor paracompact.
Since metrizability is such a desirable property for a topological space, it is common to add paracompactness to the definition of a manifold.
Paracompact manifolds have all the topological properties of metric spaces.
This is precisely the condition required to ensure that the manifold embeds in some finite-dimensional Euclidean space.
However, the converse is nearly true: a paracompact manifold is second-countable if and only if it has a countable number of connected components.
[7] By definition, every point of a locally Euclidean space has a neighborhood homeomorphic to an open subset of
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
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).
with overlapping domains U and V, there is a transition function Such a map is a homeomorphism between open subsets of
[8] A classification of 3-manifolds results from Thurston's geometrization conjecture, proven by Grigori Perelman in 2003.
More specifically, Perelman's results provide an algorithm for deciding if two three-manifolds are homeomorphic to each other.
[9] 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.
[7] The connected sum of two n-manifolds is defined by removing an open ball from each manifold and taking the quotient of the disjoint union of the resulting manifolds with boundary, with the quotient taken with regards to a homeomorphism between the boundary spheres of the removed balls.
