If I'm not mistaken, dimension can be defined easily for a topological manifold, which is actually a more fundamental structure than a differential manifold. (The former requires that chart overlaps be homeomorphisms, i.e. continuous bijections), while the latter requires that they be diffeomorphisms, i.e. smooth bijections). Smooth manifolds don't form a nice category for technical reasons, but one can think of a forgetful functor from differential manifolds to topological manifolds, and dimension being defined in the latter.
I don’t disagree that one can define dimension for a manifold (topological or differentiable). I was replying to the parent comment and so I was pointing out that dimension isn’t really well defined for a general topological space.
