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?



some questions on Lindelöf property
A space is Lindelöf iff it satisfies condition A and is metaLindelöf (every open cover has a point-countable refinement), so one could say the difference is metaLindelöfness.


