Manifold

Definition
A topological space is said to be a manifold if it satisfies all of 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 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.

Potentially weaker properties
It is most likely true that every manifold is a CW-space, i.e., it can be given the structure of a CW-complex. However, the case of dimension four is still open. See for instance this Math Stack Exchange question.

Incomparable properties
A topological space that occurs as the geometric realization of a simplicial complex is termed a polyhedron. Not every manifold is a polyhedron. For instance, the E8 manifold in 4 dimensions is not a polyhedron. Conversely, not every polyhedron is a manifold. For instance, the geometric realization of any graph with a vertex of degree more than two is not a manifold.

A manifold that is homeomorphic to the geometric realization of a simplicial complex is termed a triangulable manifold.

Metaproperties
A direct product of manifolds is again a manifold.

Any covering space of a manifold naturally gets the structure of a 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.