Find a topological space which does not have a countable basis.
Definition of basis : A collection of subsets which satisfy:
(i) union of every set equals the whole set
(ii) any element from an intersection of two subsets is contained in another subset which is itself contained in the intersection
The Attempt at a Solution
I thought I had it nailed, but the solution i came up with was actually an uncountable basis for the standard euclidean topology, which can also have countable bases. So i need to think of a space which cannot have a countable basis.
I started thinking about a space of cantor-like numbers (infinite strings of 1's and 0's) but this is still a subset of the reals! I am stuck please any hints would be a blessing.