Is topology invariant under conformal transformation? Can conformal transformation change the topology of a manifold?
In other words, if two manifolds are conformal, should they have the same topology?
If $(M,g)$ and $(N,h)$ are conformally equivalent we have a diffeomorphism $f:M\rightarrow N$ such that $f^*h$ is conformally equivalent to $g$. Since $f$ is a diffeomorphism the topology on $M$ and $N$ are the same (they have the same open sets).
I don't think they should have the same topology, but if $B_1$ is a topology base of your first manifold $S_1$, and $f$ is a conformal transformation such as $f(S_1) = S_2$ (where $S_2$ is your second manifold), my guess is that $B_2 = f(B_1)$ is a topology base of $B_1$... but I'm not sure of it