Manifold
Geometry and topology
A space in which every point has a neighbourhood resembling ordinary n-dimensional Euclidean space, for one fixed n, and which is Hausdorff and second-countable.
What is usually left out
The fixed dimension matters: a space that is two-dimensional in one region and three-dimensional in another is not a manifold. The two point-set conditions exclude pathologies such as the line with two origins and the long line, which satisfy the local condition and behave nothing like surfaces.