Defn: Suppose f : W → N where W ⊂ M and N are smooth manifolds. f is smooth if for all p ∈ W, ∃ charts (ϕ, U) and (ψ, V) and such that p ∈ U, f(p) ∈ V, f(U) ⊂ V and ψ ∘ f ∘ ϕ^{−1} is smooth.

0.) If f : W → N is smooth, then given charts (ϕ, U) and (ψ, V) such that f(U) ⊂ V, then ψ ∘ f ∘ ϕ^{−1} is smooth.

1.) If f : W → N is smooth, then f is continuous.

2.) If f : W → N is smooth, V^{open} ⊂ W, then f : V → N is smooth.

3.) f : W → N, W = ∪U_{α}^{open}, and f : U_{α} → N smooth for all α, then f : W → N is smooth.

4.) If f, g smooth, f ∘ g is smooth.

Defn: f : M → N is a diffeomorphism if f is a homeomorphism and if f and f^{−1} are smooth. M and N are diffeomorphic if there exists a diffeomorphism f : M → N.

Prop 1.2.9: Let M and N be smooth manifolds, and let {U_{α}, ϕ_{α}} be the atlas for N. Suppose f : M → N is a diffeomorphism. Then {f^{−1}(U_{α}), ϕ_{α} ∘ f} is the atlas for M.