Topological Manifold

A topological space  satisfying some separability (i.e., it is a T2-space) and countability (i.e., it is a paracompact space) conditions such that every point  has a neighborhood homeomorphic to an open set in  for some . Every smooth manifold is a topological manifold, but not necessarily vice versa. The first nonsmooth topological manifold occurs in four dimensions.

Nonparacompact manifolds are of little use in mathematics, but non-Hausdorff manifolds do occasionally arise in research (Hawking and Ellis 1975). For manifolds, Hausdorff and second countable are equivalent to Hausdorff and paracompact, and both are equivalent to the manifold being embeddable in some large-dimensional Euclidean space.

REFERENCES:

Hawking, S. W. and Ellis, G. F. R. The Large Scale Structure of Space-Time. New York: Cambridge University Press, 1975.