n. 1. a collection of objects of a set. For example, an affine manifold is merely an affine subset of a vector space. 2. (Differential topology) a topological space each point of which has a neighborhood homeomorphic to the interior of a sphere in a Euclidean space of fixed dimension; then the neighborhood of a point together with the function mapping it into Rⁿ is a chart or (local) coordinate system, and a collection of charts that cover the manifold is called an atlas. Formally, M is an n-dimensional manifold if there is a locally finite open cover, {U_λ : λ ∈ Λ}, of M such that for each λ there is a map φ_λ that maps U_λ homeomorphically onto an open subset of Rⁿ; the pair (φ_λ, U_λ) is then a chart, and the set

is an atlas for M. The pair (M, Φ) is a C^(r)-manifold if Φ is a C^(r) differential structure. Compare analytic structure.