1) This is example 5 (page 177 of Dugundji's book). Every subspace of the Sorgenfrey line is separable yet it is not second countable. I know how to prove it is not second countable. My question is: why is every subspace of the Sorgenfrey line separable? the Sorgenfrey line is separable but not metrizable, so I don't see why this follows immediately.
2) Is $\mathbb{R}^{\omega}$ a Lindelöf space? (here $\omega$ means the natural numbers). Can we simply say that $\mathbb{R}^{\omega}$ is metrizable being the countable product of a metrizable space, $\mathbb{R}$, and also separable since the countable product of separable spaces is separable so we have a separable and metrizable space, thus Lindelöf. What is another way of seeing this?
Now just for fun, what if we consider $\mathbb{R}^{\mathbb{R}}$. Is this Lindelöf?