Let $O$ be a nonempty open subset of a separable (Lindelöf) topological space $( X,\tau)$. Prove that $O$ as a subspace of $X$, is also separable (Lindelöf).
(1-separable) Given a countable dense subset of $X$, if I intersect with the open subset of $X$, we get a countable dense subset of the open subset.
(2-Lindelöf) Let $U$ be an open covering of the subspace $O$. Since all the elements of $U$ are open are open $O$, they equal the intersection of some family of open sets with $X$, call it $U'$.
I am kind of in the clouds about this proof. Please help me. Thank you! Klara