# Any Lindelöf subspace of $\omega_1^\omega$ is second countable

Show that any Lindelöf subspace of $\omega_1^\omega$ is second countable.

What I have tried: Suppose that the subspace $Y$ is Lindelöf, then $P_n(Y)$ is countable for any $n\in \omega$, and hence $Y$ is countable Lindelöf space. (see the link.) Is every countable Lindelöf space second countable? I'm not sure. I don't know how to continue.

Note that $\omega_1^\omega$ is first countable (since $\omega_1$ is), and therefore the subspace $Y$ is first-countable itself. A countable first-countable space must be second-countable.