The topology on coordinate charts of a manifold Quick question. In order to define a topological manifold on a space $M$, we have to define an atlas, or a collection of charts $(U,\phi$) such that the $U$ cover $M$ and each $\phi:U \rightarrow \mathbb{R}^{N}$ is a homemorphism.
My question is, it is my understanding that the $U$ also have a topology defined on them (in order to make continuity arguments). But, what topology exactly? If $M$ is initially endowed with a topology $\tau$, then it seems natural that the $U$ are selections from that topology and inherit the subspace topology. But must this be the case? Can we assign different underlying topologies to different $U$? 
Thanks in advance! 

The homeomorphisms must be with respect to the subspace topology on $U$ inherited from $M$; this makes $M$ into a manifold. Otherwise we would not be able to speak of $M$ alone as a manifold, but we would need to talk about $M$ together with particular topologies on particular subspaces.
