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
Contents
Definition
A topological space is said to be a manifold if it satisfies the following conditions:
- It is Hausdorff: any two distinct points can be separated by disjoint open subsets.
- It is second-countable: it has a countable basis.
- It is locally Euclidean, viz., every point has a neighborhood 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 -manifold.
Significance of three parts of the definition
Significance of local Euclideanness
Locally Euclidean is the most important property of manifolds, since this means that all the nice properties that we know about Euclidean spaces, are applicable locally. Thus, manifolds are locally contractible, locally path-connected, locally metrizable, and so on. Also, many properties of the manifold that are essentially local in nature can be proved using local Euclideanness.
Significance of Hausdorffness
If we do not assume Hausdorffness, we get pathologies like the line with two origins.
More pertinently, the important way in which we use Hausdorffness is as follows: in a Hausdorff space, any compact subset is closed. Thus, in particular, the images of closed discs of Euclidean space, inside the manifold, continue to remain closed in the whole manifold. This is crucial to applying the gluing lemma for closed subsets, for proofs like those of the fact that any connected manifold is homogeneous or that the inclusion of any point in a manifold is a cofibration.
Significance of second-countability
The assumption of second-countability can be dispensed with for a number of purposes, but is crucial for some applications. The standard example of something that is a manifold but for the second-countability assumption, is the long line.
Relation with other properties
Stronger properties
Weaker properties
- Manifold with boundary
- Closed sub-Euclidean space: For full proof, refer: Manifold implies closed sub-Euclidean
- Metrizable space
- Paracompact Hausdorff space
- Normal space
- Locally Euclidean space
- Locally contractible space
- Locally metrizable space
- Nondegenerate space: For full proof, refer: Manifold implies nondegenerate
- Compactly nondegenerate space
- Space with homology of finite type
Metaproperties
Products
This property of topological spaces is closed under taking finite products
A direct product of manifolds is again a manifold. Fill this in later
Covering spaces
This property of topological spaces is closed under passing to covering spaces; viz if a topological space has this property, so does any covering space of it
Any covering space of a manifold naturally gets the structure of a manifold.
Fiber bundles
This property of topological spaces is a fiber bundle-closed property of topological spaces: it is closed under taking fiber bundles, viz if the base space and fiber both satisfy the given property, so does the total space.
Manifold, Orientable manifold
If 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. Munkres^{More info}, Page 225, Chapter 4, Section 36 (formal definition, as definition of -manifold, where is the dimension)