By the construction of a manifold, for each point \\(s \in M\\), there exists a "chart" which is a one-to-one mapping between a neighborhood around \\(s\\) in \\(M\\) and an open subset of \\(\mathbb{R}^n\\), where \\(n\\) is the dimension of the manifold.

Footnote: it's more than just one-to-one. A manifold \\(M\\) is not just a set, but a topological space, and the charts are topological isomorphisms (aka homeomorphisms) - continuous functions with continuous inverses.

The chart establishes a strong and "smooth" correspondence between the neighborhood in \\(M\\) and the neighborhood in the parameter space \\(\mathbb{R}^n\\).

Footnote: it's more than just one-to-one. A manifold \\(M\\) is not just a set, but a topological space, and the charts are topological isomorphisms (aka homeomorphisms) - continuous functions with continuous inverses.

The chart establishes a strong and "smooth" correspondence between the neighborhood in \\(M\\) and the neighborhood in the parameter space \\(\mathbb{R}^n\\).