# Upper bound on the number of charts needed to cover a topological manifold

If $M^n$ is a compact topological manifold (not necessarily with additional structure), is there an upper bound on the number of charts needed to cover $M$ ? Does this bound depend on the dimension of $M$ ?

If you still care about this old question: An upper bound can be taken to be $n+1$ (compactness is not relevant here). I do not know if this bound can be improved. –  studiosus Mar 15 at 3:28