Diffeomorphisms
Why
We want to think about two abstract spaces as being equivalent.1
Definition
Let $X \subset \R ^n$ and $Y \subset \R ^m$. A smooth, invertible function $f: X \to Y$ is a diffeomorphism $f^{-1}$ is smooth. $X$ and $Y$ are diffeomorphic if such a function exists.
The key is the relation diffeomorphic is an equivalence relation. It is reflexive because the identity map is smooth and invertible. It is symmetric since if $f$ is a diffeomorphism from $X$ to $Y$ then $f^{-1}$ is a diffeomorphism from $Y$ to $X$. It is transitive because the composition of two smooth functions is smooth.
Differential Topology
Differential topology studies properties of $X \subset \R ^n$ which do not change under diffeomorphism.
- Future editions will include. ↩︎