Some more examples of monotonically Lindelِf and not monotonically Lindelِf spaces

A space is monotonically Lindelِf (mL) if one can assign to every open cover U a countable open refinement r ( U ) (still covering the space) so that r ( U ) refines r ( V ) whenever U refines V . Some examples of mL and non-mL spaces are considered. In particular, it is shown that the product of a mL space and the convergent sequence need not be mL, that some L-spaces are mL, and that C p ( X ) is mL only for countable X.