A Topological Space satisfying some separability (i.e., it is a Hausdorff 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 4-D.
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.
See also Hausdorff Space, Manifold, Paracompact Space, Smooth Manifold, Topological Space
Hawking, S. W. and Ellis, G. F. R. The Large Scale Structure of Space-Time.
New York: Cambridge University Press, 1975.