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-coutanble, 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 ${\varphi }_i:U_i\to V_i$. Then the transition function for these coordinate charts is defined as the map from ${\varphi }_1(U_1\cap U_2)$ to ${\varphi }_2(U_1\cap U_2)$ given by ${\varphi }_2\circ \underset{1}{\overset{-1}{\varphi }}$ (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.