prove that $n$-manifold has a countable basic

Question

$M$ is a $n$-topological manifold if it is a a topology space and each point of $M$ has a neighborhood that is homeomorphic to $\mathbb{R}^n$.

How do I prove that $n$-topological manifold has a countable basic

Thank you any help

Answer 1

With this formulation, you can't. $M$ might be the disjoint union of uncountably many copies of $\mathbb R^n$.