Productively Lindelِf and indestructibly Lindelِf spaces

There has recently been considerable interest in productively Lindelِf spaces, i.e. spaces such that their product with every Lindelِf space is Lindelِf. See e.g. [5,31,1,30,26,24,4], and work in progress by Brendle and Raghavan. Here we make several related remarks about such spaces. Indestructible Lindelِf spaces, i.e. spaces that remain Lindelِf in every countably closed forcing extension, were introduced in [28]. Their connection with topological games and selection principles was explored in [27]. We find further connections here.