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?