I am trying to learn differential geometry on my own, and I'm finally getting serious about learning the subject. I am using several books which I supplement with material I find on the internet. I have found some excellent lecture notes on differential geometry written by Dmitri Zaisev. I like his discussion of tangent vectors from several viewpoints. My question concerns a definition that I have found in those notes. The definition of a submanifold is given as: A subset S ⊂ ℝn is an m-dimensional submanifold of ℝn (m < n) of class Ck if ∀ p in S ∃ an open neighborhood Vp ⊂ ℝn, an open set Wp ⊂ ℝm, and a homeomorphism fp: Wp → Vp ∩ S which is Ck and regular in the sense that ∀ a ∈ Wp, the differential dafp: ℝm → ℝn is injective. I need help pulling this definition apart. In particular, I am perplexed as to the part the set Wp plays. If we need an open set in S, why isn't Vp ∩ S an open set in S? Why do we need fp to map from Wp to Vp ∩ S because it seems to me that they are both sets contained in S.