Abstract:A space is monotonically Lindel\"of (mL) if one can assign to every open cover $\Cal U$ a countable open refinement $r(\Cal U)$ so that $r(\Cal U)$ refines $r(\Cal V)$ whenever $\Cal U$ refines $\Cal V$. We show that some countable spaces are not mL, and that, assuming CH, there are countable mL spaces that are not second countable.
Keywords: Lindel\"of, monotonically Lindel\"of, tower, the countable fan space, Pixley-Roy space
AMS Subject Classification: 54D20