Not sure what you mean here. But it is indeed true that any open set in R can be written as union of open balls. This is essentially lemma 13.1 of page 80... It can even be shown that this union is countable.
OK, actually it's not important, let's ignore it.
So, the point is - from my original basis C, I ended up with its countable subcollection C'', which is a basis for a topology coarser than the one generated by C. Actually this is quite obvious, since C'' is obtained by removing sets from C.
I'll try to solve this tomorrow, gotta get some sleep now.
Thanks for the help!