## Definition

A **differential manifold** or **smooth manifold** is the following data:

- A topological space
- An atlas of coordinate charts on to (in other words an open cover of with homeomorphisms from each member of the open cover to open sets in

satisfying the compatibility condition: the transition function between any two coordinate charts of the atlas is a diffeomomorphism of open subsets of .

By diffeomorphism, we here mean a map with a inverse.

upto the following equivalence:

Two atlases of coordinate charts on a topological space define the same differential manifold structure if given any coordinate chart in one and any coordinate chart in the other, the transition function between them is a diffeomorphism.