Properties related to star countability and star finiteness

Two star properties recently studied by Song and a generalization of countable compactness called weak star finiteness by Song and previously, 1-cl-starcompactness by Matveev and Ikenaga, are studied. We show that two of these properties coincide with feeble Lindelöfness and feeble compactness respectively in the class of spaces with a dense set of isolated points. Preservation of one of these properties under products by compact and sequentially compact spaces is also considered.