Here is a theorem which said:
Every locally metrizable linearly lindelöf space $X$ is a separable metrizable space (hence, lindelöf).
Somebody said, it is enough to show that $X$ is hereditarily lindelöf. I cannot follow him. Could somebody help me?
If I may ask more, he also said, if $X$ is not hereditarily, then $X$ contains a subspace $Y$ such that $|Y|=\omega_1$ and each uncountable subspace $Z$ of $Y$ is not lindelöf. How to show such subspace $Y$ exists? Thanks ahead:)