# 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! im stuck please any hints would be a blessing.

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.