The product of metrizable space and one point lindelofication of the discrete space Let $X$ be the one point lindefication of a discret space of cardinality $\omega_1$ and $Y$ is any metrizable space. Is $X \times Y$ always Lindelöf?
If I may ask more, does a metric space $Y$ with $e(Y)=\omega$ which is not lindelöf exist?

No. Just start with a metric space $Y$ that isn’t Lindelöf: then $\{0\}\times Y$ is a closed subset of $X\times Y$ that isn’t Lindelöf, so $X\times Y$ can’t be Lindelöf, either.
