Manifold

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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 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 m, then we call the manifold a m-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

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 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. MunkresMore info, Page 225, Chapter 4, Section 36 (formal definition, as definition of m-manifold, where m is the dimension)