On the generalized Lindelöf property (Q1852359)

The authors define several generalizations of Lindelöf spaces: (1) A space \(X\) is para-Lindelöf if every open cover has a locally countable open refinement, (2) \(X\) is almost Lindelöf if every open cover has a countable subcollection whose union is dense in \(X\) (analogous definitions are given for almost compact and almost paracompact), and (3) \(X\) is almost para-Lindelöf if every open cover has a locally countable open refinement whose union is dense in \(X\). Examples are given of spaces which satisfy the weaker property but not the stronger in each case, and almost Lindelöfness and almost paracompactness are shown to be independent of each other. Invariance of these properties under certain mappings is discussed in one section, and in another section these properties are studied in particular spaces.