A manifold is a topological space that locally resembles euclidean space near each point. an n-dimensional manifold is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of n-dimensional Euclidean space.
A d-dimensional space is said to be a manifold if and only if at each point there exists a neighborhood that is homeomorphic to d-dimensional Euclidean space, R d .