A topological space is said to be a topological manifold or simply manifold) of dimension if it satisfies the following conditions:
- It is Hausdorff
- It is second-countable, viz it has a countable basis of open sets
- Every point in the space has a neighbourhood that is homeomorphic to an open set in . Such a neighbourhood is termed a coordinate neighbourhood and the homoemorphism is termed a coordinate chart.
Relation with other structures
Coordinate charts and atlases
A topological atlas on a topological manifold is the following data:
- An open cover of the manifold
- For each member of the open cover, ahomoemorphism from that open set to an open set in . Since this homeomorphism associates Euclidean coordinates to points in that open set, it is termed a coordinate chart
Transition function between two coordinate charts
Suppose are open subsets of , and open subsets of , with homeomorphisms . Then the transition function for these coordinate charts is defined as the map from to given by (note that both these maps make sense in the area we are defining).
Intuitively, the transition function is a way to pass from one coordinate chart to another, for those points for which both coordinate charts are valid.