Topological 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 $\mathbb {R} ^{n}$ . 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 $\mathbb {R} ^{n}$ . 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_1, U_2} are open subsets of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_1, V_2} open subsets of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R^n} , with homeomorphisms Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_i: U_i \to V_i} . Then the transition function for these coordinate charts is defined as the map from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_1(U_1 \cap U_2)} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_2(U_1 \cap U_2)} given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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.