A Hausdorff topological space with an n-dimensional atlas of charts for some integer n. The integer n is called the dimension of the manifold. If M is a Hausdorff space, an n-chart on M is a pair (U, ϕ) with U an open subset of M and ϕ a homeomorphism of U onto an open subset ϕ(U) of n-dimensional euclidean space Rⁿ. An n-dimensional atlas for M is a system of n-charts (U, ϕ) such that the union of the sets U is all of M.