Classification of Manifolds
A 0-manifold is just a discrete space. Such spaces are classified by their cardinality. Every discrete space is paracompact. A discrete space is second-countable if and only if it is countable.
Every nonempty, paracompact, connected 1-manifold is homeomorphic either to R or the circle. The unconnected ones are just disjoint unions of these.
Every nonempty, compact, connected 2-manifold (or surface) is homeomorphic to the sphere, a connected sum of tori, or a connected sum of projective planes. See the classification theorem for surfaces for more details.
A classification of 3-manifolds results from Thurston's geometrization conjecture whose proof was sketched by Grigori Perelman. The details have been provided by other members of the mathematical community.
The full classification of n-manifolds for n greater than three is known to be impossible; it is at least as hard as the word problem in group theory, which is known to be algorithmically undecidable. In fact, there is no algorithm for deciding whether a given manifold is simply connected. There is, however, a classification of simply connected manifolds of dimension ≥ 5.
Read more about this topic: Topological Manifold