I don't think I understand what exactly a manifold is until I watched Milnor's video:
"Def $X$ is a smooth manifold of dim $m$ if each $x\in X$ has a neighborhood diffeomorphic to a open subset of $\mathbb R^m$."
So changing "diffeo" to "homeo" we get the definition of manifold?
By this definition, a manifold of dim $m$ must be also a manifold of dim $m+1$?
We can also change the $\mathbb R^m$ to any space like Banach to make a Banach manifold? But $\mathbb R^m$ is field and Banach is a space, how could you freely interchange the field and space?