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Lindelöf; monotonically Lindelöf; tower; the countable fan space; Pixley-Roy space
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.
[1] Baumgartner J.E.: Chains and antichains in $\Cal P(ømega)$. J. Symbolic Logic 45 (1980), 1 85-92. DOI 10.2307/2273356 | MR 0560227
[2] Bennett H., Lutzer D., Matveev M.: The monotone Lindelöf property and separability in ordered spaces. Topology Appl. 151 (2005), 180-186. DOI 10.1016/j.topol.2004.05.015 | MR 2139751 | Zbl 1069.54021
[3] van Douwen E.K.: Integers in topology. Handbook of Set-theoretic Topology, K. Kunen and J. E. Vaughan, eds., Elsevier Sci. Pub. B.V., 1984, pp.111-168.
[4] van Douwen E.K.: The Pixley-Roy topology on spaces of subsets. Set-theoretic topology (Papers, Inst. Medicine and Math., Ohio Univ., Athens, Ohio, 1975-1976), 113-134, Academic Press, New York, 1977. MR 0440489 | Zbl 0372.54006
[5] van Douwen E.K., Kunen K.: L-spaces and S-spaces in $\Cal P(ømega)$. Topology Appl. 14 (1982), 2 143-149. DOI 10.1016/0166-8641(82)90064-5 | MR 0667660
[6] Engelking R.: General Topology. Heldermann Verlag, Berlin, Sigma Series in Pure Mathematics, 6, 1989. MR 1039321 | Zbl 0684.54001
[7] Gruenhage G.: Generalized metric spaces. in: Handbook of Set-theoretic Topology, K. Kunen and J. E. Vaughan, eds., Elsevier Sci. Pub. B.V., 1984, pp.423-501. MR 0776629 | Zbl 0794.54034
[8] Junnila H.J.K., Künzi H.-P.A.: Ortho-bases and monotonic properties. Proc. Amer. Math. Soc. 119 (1993), 4 1335-1345. DOI 10.1090/S0002-9939-1993-1165056-6 | MR 1165056
[9] Levy R., Matveev M.: Some examples of monotonically Lindelöf and not monotonically Lindelöf spaces. Topology Appl. 154 (2007), 2333-2343. MR 2328016
[10] Matveev M.: A monotonically Lindelöf space need not be monotonically normal. preprint, 1994.
[11] Todorčević S.: Partition Problems in Topology. Contemporary Mathematics, 84, American Mathematical Society, Providence, Rhode Island, 1989. DOI 10.1090/conm/084 | MR 0980949
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