In this article, two properties are mentioned at page 153:
property $\gamma$: If $\mathcal U$ is an open $\omega$-cover of $X$, then there is a sequence $G_n \in \mathcal U$, with $\underline {Lim} G_n = X$.
property $\delta$: If $\mathcal U$ is an open $\omega$-cover of $X$, then, $X \in L(\mathcal U)$
My question is: Can anyone think of a topological space $X$ which satisfies property $\delta$ and does not satisfy property $\gamma$?
Thank you!
Remarks: 1. If $\langle A_n : n \in \omega \rangle$ is a sequence of subsets of a set $X$, $$ \underline{Lim} A_n = \{ x \in X : \exists n_0 \in \omega \forall n \geq n_0, x \in A_n \} $$ If $\mathcal A$ is a family of subsets of a set $X$, then, $L(\mathcal A)$ denotes the smallest family of subsets of $X$ containing $\mathcal A$ and closed under $\underline{Lim}$.
2.An open cover $\mathcal U$ of a topological space $(X,\tau)$ is called an $\omega$-cover if $X \notin U$ and for every finite set $K \subset X$, there is a set $U \in \mathcal U$ such that $K \subset U$.