Are countable topological spaces second-countable?

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.

Take a free ultrafilter on $\mathbb{N}$ and add $\emptyset$ to it to make it a topology. Standard facts on ultrafilters tell us that this is not second countable.

Another advanced example :let $x \in \omega^\ast$ and let $X= \omega \cup \{x\}$ in the subspace topology.

Or let $D$ be a countable dense subset of $\{0,1\}^{\Bbb R}$ in the product topology. I wrote extensively on such spaces here.