As a continuation to this question:
An $\omega$-cover, is an open cover $\mathcal U$ of $X$, such that, $X \notin \mathcal U$, and for every finite set $F \subset X$, there exists an open set $U \in \mathcal U$, such that, $F \subset U$.
Let $\mathcal U$, be an $\omega$-cover of $X$, such that, $X \in L(\mathcal U)$. Does this imply that $\mathcal U$, is also a $\gamma$-cover?
(This question of mine is a consequence of trying to prove that $\delta \Rightarrow \epsilon$ in page 153 here)
Thank you!