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

Question

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.

Thanks ahead:)

Answer 1

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.

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The Arens-Fort space is an example of a countable (hence Lindelöf) space which is not second-countable (or even first-countable).