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

• Differential manifold
• Measured manifold

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 $U_{1},U_{2}$ are open subsets of $X$, and $V_{1},V_{2}$ open subsets of $\mathbb {R} ^{n}$, with homeomorphisms $\phi _{i}:U_{i}\to V_{i}$. Then the transition function for these coordinate charts is defined as the map from Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \phi _{1}(U_{1}\cap U_{2})} to Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \phi _{2}(U_{1}\cap U_{2})} given by Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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.