I know that $\Bbb R_l$ is first countable and separable but not second countable.
But I had earlier tried to prove that first countable separable spaces are second countable. So I wrote a proof. Now I know a counterexample, but still I can not find a mistake in my proof.
My "proof" for the wrong fact:
First countable separable spaces are second countable.
Proof : As we have separable space there is countable dense set, say $A$.
Now for each point in $A$ we can find a countable basis.
Countable unions of countable set is countable.
Claim: Set formed by taking the union of all countable basis at the dense set is required basis.
Take any $x\in X$. Choose any neighborhood of $x$. Now as $A$ is dense, there exists some point in $A$ that is in the given neighborhood and so choose one of the countable basis element that contained in that neighborhood.
So we are done.
Actually, I was thinking maybe the last statement has a problem. Is it so? Why not countable basis element at that point should not contain in given neighborhood?
Any help will be appreciated.