Topological manifold

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The article on this topic in the Topology Wiki can be found at: manifold

Definition

A topological space is said to be a topological manifold or simply manifold) of dimension n 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 \R^n. Such a neighbourhood is termed a coordinate neighbourhood and the homoemorphism is termed a coordinate chart.

Relation with other structures

Stronger structures

Facts

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 \R^n. 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 U_1, U_2 are open subsets of X, and V_1, V_2 open subsets of \R^n, with homeomorphisms \phi_i: U_i \to V_i. Then the transition function for these coordinate charts is defined as the map from \phi_1(U_1 \cap U_2) to \phi_2(U_1 \cap U_2) given by \phi_2 \circ \phi_1^{-1} (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.