Is it possible to define a Borel $\sigma$-algebra on a topological space which is not second countable, i.e. one which does not have a countable base?
I am trying to learn measure theory and my intuition says that this is NOT possible, because if the space is not second countable, then there are "too many" open sets to be able to get closure taking countable unions (etc.) of open sets.
(The book by Bogachev (and other books) did not help me understand this any better, which is why I am asking here.)