The topology on coordinate charts of a manifold

Question

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!

Answer 1

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.

Answer 2

Being a 'topological manifold' is a property that a topological space can have, it does not come with extra structure.

This answers two things: First, you indeed have to start with a topological space $(X,\tau)$. Second: Every other property is to be understood with respect to this topology $\tau$. That is, a chart is a homeomorphism (i.e. a map continuous with respect to $\tau$ which also has an inverse which is continuous with respect to $\tau)$ $\phi\colon U\rightarrow \mathbb{R}^n$ for some $U\in \tau$.

That is, $(X,\tau)$ is a topological manifold, if it is (Hausdorff, $2$nd countable and) locally euclidean, i.e. if every point $x$ is contained in the domain $U$ of a chart.