# scattered line is not metrizable

How to prove: scattered line is not metrizable?

I proved that it is not second countable, not separable. It is normal. Urysohn metrization theorem wont help as it is not separable. What property of metric space is it not satisfying?

HINT: Prove that $\Bbb Q$ is a closed set that is not a $G_\delta$-set: in a metric space every closed set is a $G_\delta$-set. You’ll want to use the Baire category theorem in $\Bbb R$. (For future reference, you may find it useful to know that this space is also called the Michael line.)