Dimension of topology manifold

Question

In the 3 page of Jurgen Jost's Riemannian Geometry and Geometric Analysis .Why it is harder in topology manifold than differentiable manifold ? I think it is easy in differentiable manifold because the Jacobian of chart transition is not zero, so every chart has same dimension . But in topology manifold ,all the open sets are open cover , they must have same dimension, so the topology manifold has same dimension everywhere.

Answer 1

To prove that the dimension of a topological manifold is well-defined, you have to prove that $\mathbb{R}^n$ is not locally homeomorphic to $\mathbb{R}^m$ for $m\neq n$. This is not at all obvious, and is definitely harder than the simple linear algebra (i.e. vector spaces have well-defined dimension) you can do to show that $\mathbb{R}^n$ is not locally diffeomorphic to $\mathbb{R}^m$.