Recall that a space $X$ is metaLindelof if every open cover of $X$ has a point-countable open refinement. A space $X$ is metacompact if every open cover of $X$ has a point-finite open refinement.
I have the following two questions, because the questons are similar, I put together here:
Is there a $\sigma$-metaLindelof space which is a not metaLindelof?
Is there a $\sigma$-metacompact space which is not metacompact?
Thanks for your help.