Are countable spaces (i.e. $\mathbb{N}$ with any topology) second-countable? A countable space can have at most $2^\omega$ open subsets which suggests that a counterexample may exist. On the other hand both discrete and anti-discrete (or more generally with countable topology) spaces are second-countable. Also note that obviously a countable space is separable. So if it is additionally metrizable then it is second-countable.
But I couldn't prove that in general. Or is there a counterexample?
Best Answer
Consider $ω$ many convergent sequences, and glue their limits. The resulting space won't have countable base at the common limit point.
Also note that a countable space is second-countable if and only if it is first-countable.