Topological manifold
From Freepedia
| This article is in need of attention. You can help Wikipedia by editing it into a better article. Please also consider changing this notice to be more specific. |
A topological manifold is a manifold that is glued together from Euclidean spaces. Euclidean spaces are the simplest examples of topological manifolds. More precisely a topological manifold is a topological space that locally looks like an Euclidean space.
Contents |
Charts and transition maps
- REDIRECT Template:Main
A K-chart at p is a homeomorphism from an open neighbourhood of p to K. Instead of saying "there is a K-chart at p", you can say "at p there is a K-chart". If at p there are two K-charts, then by restricting them to the intersection of their domains we can compose the inverse of one with the other to form a transition map from K to itself. Thus all transition maps are homeomorphisms.
Topological manifolds
Topological manifold without boundary
The prototypical example of a topological manifold without boundary is Euclidean space. A general manifold without boundary looks locally, as a topological space, like Euclidean space. This is formalized by requiring that a manifold without boundary is a non-empty topological space in which every point has an open neighbourhood homeomorphic to (an open subset of) Rn (Euclidean n-space). Another way of saying this, using charts, is that a manifold without boundary is a non-empty topological space in which at every point there is an Rn-chart.
Topological manifold with boundary
More generally it is possible to allow a topological manifold to have a boundary. The prototypical example of a topological manifold with boundary is the Euclidean closed half-space. Most points in Euclidean closed half-space, those not on the boundary, have a neighbourhood homeomorphic to Euclidean space in addition to having a neighbourhood homeomorphic to Euclidean closed half-space, but the points on the boundary only have neighbourhoods homeomorphic to Euclidean closed half-space and not to Euclidean space. Thus we need to allow for two kinds of points in our topological manifold with boundary: points in the interior and points in the boundary. Points in the interior will, as before, have neighbourhoods homeomorphic to Euclidean space, but may also have neighbourhoods homeomorphic to Euclidean closed half-space. Points in the boundary will have neighbourhoods homeomorphic to Euclidean closed half-space. Thus a topological manifold with boundary is a non-empty topological space in which at each point there is an Rn-chart or an [0,∞)×Rn−1-chart. The set of points at which there are only [0,∞)×Rn−1-charts is called the boundary and its complement is called the interior. The interior is always non-empty and is a topological n-manifold without boundary. If the boundary is non-empty then it is a topological (n-1)-manifold without boundary. If the boundary is empty, then we regain the definition of a topological manifold without boundary.
Sheaf of continuous functions
The continuous real-valued functions form a sheaf. An alternative definition of a topological manifold is a topological space with a sheaf of continuous functions locally isomorphic to Euclidean space with its sheaf of continuous functions.
Properties
A manifold with empty boundary is said to be closed if it is compact, and open if it is not compact.
Manifolds inherit many of the local properties of Euclidean space. In particular, they are locally path-connected, locally compact and locally metrizable. Being locally compact Hausdorff spaces they are necessarily Tychonoff spaces. Requiring a manifold to be Hausdorff may seem strange; it is tempting to think that being locally homeomorphic to a Euclidean space implies being a Hausdorff space. A counterexample is created by deleting zero from the real line and replacing it with two points, an open neighborhood of either of which includes all nonzero numbers in some open interval centered at zero. This construction, called the real line with two origins is not Hausdorff, because the two origins cannot be separated.
A topological space is said to be homogeneous if its homeomorphism group acts transitively on it. Every connected manifold without boundary is homogeneous, but manifolds with nonempty boundary are not homogeneous.
It can be shown that a manifold is metrizable if and only if it is paracompact. Non-paracompact manifolds (such as the long line) are generally regarded as pathological, so it's common to add paracompactness to the definition of an n-manifold. Sometimes n-manifolds are defined to be second-countable, which is precisely the condition required to ensure that the manifold embeds in some finite-dimensional Euclidean space. Note that every compact manifold is second-countable, and every second-countable manifold is paracompact.
Subtypes
Piecewise linear manifold
Merge from piecewise linear
Differentiable manifold
Technical details
Topological manifolds are usually required to be Hausdorff and second-countable.
Another generalization of manifold allows one to omit the requirement that a manifold be Hausdorff. It still must be second-countable and locally Euclidean, however. Such spaces are called non-Hausdorff manifolds and are used in the study of codimension-1 foliations.
The Hausdorff assumption
Line with two origins
Here we will present the example of a space which satisfies all the assumptions of a topological manifold except that it is not a Hausdorff space. Take two copies of R, write them as
- <math>\mathbf{R}\times\{0\}</math> and <math>\mathbf{R}\times\{1\}</math>,
and define an equivalence relation by
- <math>(x,0) \sim (x,1)</math> if <math>x \neq 0</math>.
The quotient space L obtained from this equivalence relation is a space like the real line, except two points "occupy" the origin. In particular, they cannot be separated by disjoint open sets, so L is non-Hausdorff. It is a topological manifold, however not a Hausdorff topological manifold.
Often, topological manifolds are defined to be Hausdorff, so examples such as this are excluded.
