I have several questions on Lindelöf property.
If every point countable open cover of $X$ has a countable subcover (Condition A), does $X$ have Lindelöf property? How far is having Condition A from Lindelöf property?
A space $X$ is called $\omega_1$-Lindelöf if every $\omega_1$-sized open cover of $X$ contains a countable subcover.
Can every $\omega_1$-Lindelöf space with Condition A be Lindelöf?
A space $X$ is called discretely Lindelöf if the closure of every discrete subspace of $X$ is Lindelöf.
Can every discretely Lindelöf space with Condition A be Lindelöf?