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.
