Let $X$ be a Lindelöf topological space. Does this imply that every $\omega$-cover has a countable subcover which is also an $\omega$-cover? if not, is there an example of a topological Lindelöf space with an $\omega$-cover for which there isn't a countable sub $\omega$-cover?
We say that an open cover $\mathcal{U}$ of $X$ is an $\omega$-cover, if $X \notin \mathcal{U}$ and for any finite set $A \subset X$ there is a $U \in \mathcal{U}$ such that $A \subseteq U$.
Thank you!