Find a topological space which does not have a countable basis

[doodlepin]

Homework Statement

Find a topological space which does not have a countable basis.

Homework Equations

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.

 

[ystael]

You have complete freedom to choose both the set and the topology you give it. So one strategy you could try is to make your underlying set very large -- say, a large number of copies of some familiar space. Alternatively you could try starting with a familiar set like $$\mathbb{R}$$, but giving it very many open sets, many more than the standard topology has.