Geodesic
On monotone Lindelöfness of countable spaces
Commentationes Mathematicae Universitatis Carolinae, Tome 49 (2008) no. 1, pp. 155-161.

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A space is monotonically Lindelöf (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.
Classification : 54D20
Keywords: Lindelöf; monotonically Lindelöf; tower; the countable fan space; Pixley-Roy space 
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Levy, Ronnie; Matveev, Mikhail. On monotone Lindelöfness of countable spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 49 (2008) no. 1, pp. 155-161. http://geodesic.mathdoc.fr/item/CMUC_2008__49_1_a14/