Choice of chart and atlas

Question

Manifold is second countable space. Should charts in atlas also be countable? I don't think it has to be. But somehow the second countable condition enforces me to think like that..

Answer 1

There can be a lot of uncountable atlases, but you can always choose a countable atlas to work with, since every second-countable space is Lindelöf. The domains of the charts in the atlas form a cover of the manifold, so you can extract a countable subcover. These charts will give you a countable atlas.

Answer 2

Since basis is countable there must be at most the "continuum number" open sets:

For each open subset we associate the set containing for elements basis open sets that are contained in it. This association is a bijection. Therefore cardinality is no more than the number of subsets of $\mathbb{N}$ which is the size of the continuum.

This is definetely not too huge, since even $\mathbb{R}$ has that many opens!

PS Don't worry to much about the axiom. It is just there to ensure you that manifolds are not too big. You can work out a lot without assuming it. But some things like the Whitney's theorem of course not.