Most of the differential topology book describe a manifold that is a subset of $\mathbb{R}^n$ and without describing the smooth structure, they talk about their properties as smooth manifolds, such as a square which is mentioned in a question in Lee's book on p75.
What is the "natural" smooth structure on the subsets of $\mathbb{R}^n$ that are topological manifolds ? Can you explicitly specify it ?
For example, if we were talking about the "natural" topology on the subsets of $\mathbb{R}^n$, it would be the topology generated by the basis whose elements are the intersection of open balls with that subset; so now the topic is about smooth structure on those topological manifolds.