On properties related to star countability (Q1663849)

It is shown that a Hausdorff metacompact (metaLindelöf) weakly star finite (countable) space is almost compact hence feebly compact (feebly Lindelöf) and that a Hausdorff weakly star countable space with rank 4-diagonal has cardinality at most \(\mathfrak c\). Some examples are also presented, such as a normal, weakly star countable space that is neither almost star countable nor star Lindelöf, with a strengthening under \(\mathsf{MA}+\neg\mathsf{CH}\) of the former situation to a space which is also Moore.