Let $\omega(X)=\min\{|\mathcal{B}|:\mathcal{B} \mbox{ is a base of the topology of } X\}$. I want to make sure whether the following statement is true : $\omega(X)=\aleph_0 $ iff $X$ is second countable.
By the definition of the weight of space $X$, if $\omega(X)=\aleph_0 $ then $X$ is second countable. I am not sure whether the converse is also correct. If not, please give me a counterexample.