Some more examples of monotonically Lindelof and not monotonically Lindelof spaces

A space is monotonically Lindelof (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. (c) 2007 Elsevier B.V. All rights reserved.