Topological Manifold - Coordinate Charts

Coordinate Charts

By definition, every point of a locally Euclidean space has a neighborhood homeomorphic to an open subset of Rn. Such neighborhoods are called Euclidean neighborhoods. It follows from invariance of domain that Euclidean neighborhoods are always open sets. One can always find Euclidean neighborhoods that are homeomorphic to "nice" open sets in Rn. Indeed, a space M is locally Euclidean if and only if either of the following equivalent conditions holds:

• every point of M has a neighborhood homeomorphic to an open ball in Rn.
• every point of M has a neighborhood homeomorphic to Rn itself.

A Euclidean neighborhood homeomorphic to an open ball in Rn is called a Euclidean ball. Euclidean balls form a basis for the topology of a locally Euclidean space.

For any Euclidean neighborhood U a homeomorphism φ : U → φ(U) ⊂ Rn is called a coordinate chart on U (although the word chart is frequently used to refer to the domain or range of such a map). A space M is locally Euclidean if and only if it can be covered by Euclidean neighborhoods. A set of Euclidean neighborhoods that cover M, together with their coordinate charts, is called an atlas on M. (The terminology comes from an analogy with cartography whereby a spherical globe can be described by an atlas of flat maps or charts).

Given two charts φ and ψ with overlapping domains U and V there is a transition function

ψφ−1 : φ(U ∩ V) → ψ(U ∩ V).

Such a map is a homeomorphism between open subsets of Rn. That is, coordinate charts agree on overlaps up to homeomorphism. Different types of manifolds can be defined by placing restrictions on types of transition maps allowed. For example, for differentiable manifolds the transition maps are required to be diffeomorphisms.