Timeline for Why do we require manifolds to be a topological space?

4 events

when toggle format	what		by	license	comment
Mar 31, 2014 at 0:33	vote	accept	Hunter
Mar 31, 2014 at 2:09
Mar 31, 2014 at 0:30	comment	added	Robin Ekman		Actually in physics we usually require diffeomorphism, because we want to talk about derivatives. So we require that the change-of-coordinate functions are smooth. Then we define $f : M\to \mathbb R$ to be smooth if $f \circ \varphi^{-1}$ is smooth. Here $\varphi : M \to \mathbb R^n$ is a coordinate function. A diffeomorphism is a homeomorpism that is smooth. With this definition the manifold becomes locally diffeomorphic to Euclidean space. Note that we had to define what it means to be differentiable on a manifold using the coordinate functions.
Mar 31, 2014 at 0:13	comment	added	Hunter		Ok, thanks, that makes sense. In physics we (always?) assume that the manifolds we are working with have a homeomorphism from the $U_i$ to $\mathbb{R}^n$. Are there also physical theories that don't have this condition? What would happen to, say, general relativity if the spaces are diffeomorphic to each other, instead of homeomorphic?
Mar 31, 2014 at 0:08	history	answered	Robin Ekman	CC BY-SA 3.0