# Does a metric Lindelöf space have a countable basis?

I want to prove that every metric space which is Lindelöf has a countable basis. First I tried to show that a countable cover, which exists by the Lindelöf property, is a countable basis, but for the second property of basis, which is about the intersection of two basis elements, I have no idea.

Since $X$ is a Lindelöf space, for every $n$ there exist countably many $x_{n,j}$ with $$\bigcup_jB(x_{n,j},1/n)=X.$$So $X$ is separable; in fact $\{x_{n,j}:n,j\in\Bbb N\}$ is dense, and hence $$\{B((x_{n,j},1/k):n,j,k\in\Bbb N\}$$is a basis.