A manifold \(M\) of dimension \(n\) is a second countable Hausdorff space which is locally Euclidean of dimension \(n\) . We call such manifolds also topological manifolds if we want to distinguish them from other manifolds like smooth and Riemannian ones.
Remarks
- The notion of dimension makes sense because of the invariance of dimensions .
- A manifold should be Hausdorff to be “fine” enough and second countability ensures not too many open sets. However, some authors consider other countability properties like first countability , separability or the Lindelöf property .
- \(n\)-manifolds with \(n\le 2\) are fully classified up to homeomorphism.
- \(3\)-manifolds were finally classified by a result of Perelman (2006). He rejected the Fields medal for that result.
- Every manifold has a basis of coordinate balls.
- Every manifold is locally path-connected.