Some questions relating topology and manifolds

[Hymne]

Hello there!
I just started reading Topological manifolds by John Lee and got one questions regarding the material.
I am thankful for any advice or answer!

The criteria for being a topological manifold is that the space is second countantable ( = there exists a countable neighborhood basis?).
I can't see why R^n fulfills this criteria.. the neighborhood basis for q is all the open sets that contain q. And if we view these as all the open balls with a radius varying on the reell line, these are not countable due to the incountability of R.. right?

In which way is R^n second countable?

 

[micromass]

Take only the balls with rational centers. Thus your basis should be
\mathcal{B}=\{B(q,1/n)~\vert~q\in \mathbb{Q},~n\in \mathbb{N}_0\}

In fact, if your space in metrizable, then the space is second countable if and only if it is separable. And \mathbb{R} is separable, since it contains \mathbb{Q} as countable dense subset...

 

[Bacle]

Just to pick up some low-hanging fruit from Micromass' response, you can see how
every metric space is also 1st-countable, by using { B(q,1/n): n in Z+ }.

 

[mathwonk]

this is exactly why in calculus, in order to show "for every epsilon there is a delta", it suffices to take epsilon equal to 1/n for all positive integers n. I.e. there are uncountably many real epsilons, but it suffices to look only at the countable set of 1/n's