Definition of manifold which are subset of euclidean space

Question

According to Guillemin and Pollack "X(which is a subset of R^n)is a k-dimensional manifold if it is locally diffeomorphic to Rk , meaning that each point x possesses a neighborhood V in X which is diffeomorphie to an open set U of R^k". My question is why does this definition not contradicting the fact the diffeomorphism is only possible between manifold of same dimension?

Answer 1

I think your confusion is around the $k$ vs. $n$. If so, I think the point is that the dimension of a manifold is not the dimension of the Euclidean space of which it is a subset (if it even is a subset of some Euclidean space).

For example, consider the Klein bottle $K$. $K$ can be viewed as a subset of $\mathbb{R}^4$, but is $2$-dimensional, and cannot be viewed as a subset of $\mathbb{R}^3$ (let alone $\mathbb{R}^2$). The number $4$ (= dimension of smallest Euclidean space into which $K$ embeds) tells you something about the manifold $K$, but it is not the dimension.

Put another way: If I have $K$ as a subspace of $\mathbb{R}^4$, then it is locally homeomorphic to $\mathbb{R}^2$; but this in no way implies that $\mathbb{R}^4$ is locally homeomorphic to $\mathbb{R}^2$.