Products of star-Lindelof and related spaces

A space is star-Lindelof if for every open cover U, the cover $\{St(x; U) : x\in X\}$ has a countable subcover. A space X is centered-LindelÁof (linked-Lindelof) if every open cover has a $\sigma$-centered ($\sigma$-linked) subcover. We survey and generalize known results on finite products of star-Lindelof spaces and obtain some new. Then we consider infinite products and prove that, for a regular space X, $X^\tau$ is linked- (or centered-, or star-) LindelÁof for every cardinal $\tau$ $X^\tau$ is countably compact for every cardinal $\tau$. We observe that non-trivial box products are never linked-Lindelof .