On monotone Lindelöfness of countable spaces

Ronnie Levy; Mikhail Matveev

Commentationes Mathematicae Universitatis Carolinae (2008)

  • Volume: 49, Issue: 1, page 155-161
  • ISSN: 0010-2628

Abstract

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A space is monotonically Lindelöf (mL) if one can assign to every open cover 𝒰 a countable open refinement r ( 𝒰 ) so that r ( 𝒰 ) refines r ( 𝒱 ) whenever 𝒰 refines 𝒱 . We show that some countable spaces are not mL, and that, assuming CH, there are countable mL spaces that are not second countable.

How to cite

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Levy, Ronnie, and Matveev, Mikhail. "On monotone Lindelöfness of countable spaces." Commentationes Mathematicae Universitatis Carolinae 49.1 (2008): 155-161. <http://eudml.org/doc/250472>.

@article{Levy2008,
abstract = {A space is monotonically Lindelöf (mL) if one can assign to every open cover $\mathcal \{U\}$ a countable open refinement $r(\mathcal \{U\})$ so that $r(\mathcal \{U\})$ refines $r(\mathcal \{V\})$ whenever $\mathcal \{U\}$ refines $\mathcal \{V\}$. We show that some countable spaces are not mL, and that, assuming CH, there are countable mL spaces that are not second countable.},
author = {Levy, Ronnie, Matveev, Mikhail},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Lindelöf; monotonically Lindelöf; tower; the countable fan space; Pixley-Roy space; Lindelöf; monotonically Lindelöf; tower; the countable fan space; Pixley-Roy space},
language = {eng},
number = {1},
pages = {155-161},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On monotone Lindelöfness of countable spaces},
url = {http://eudml.org/doc/250472},
volume = {49},
year = {2008},
}

TY - JOUR
AU - Levy, Ronnie
AU - Matveev, Mikhail
TI - On monotone Lindelöfness of countable spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2008
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 49
IS - 1
SP - 155
EP - 161
AB - A space is monotonically Lindelöf (mL) if one can assign to every open cover $\mathcal {U}$ a countable open refinement $r(\mathcal {U})$ so that $r(\mathcal {U})$ refines $r(\mathcal {V})$ whenever $\mathcal {U}$ refines $\mathcal {V}$. We show that some countable spaces are not mL, and that, assuming CH, there are countable mL spaces that are not second countable.
LA - eng
KW - Lindelöf; monotonically Lindelöf; tower; the countable fan space; Pixley-Roy space; Lindelöf; monotonically Lindelöf; tower; the countable fan space; Pixley-Roy space
UR - http://eudml.org/doc/250472
ER -

References

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  8. Junnila H.J.K., Künzi H.-P.A., 10.1090/S0002-9939-1993-1165056-6, Proc. Amer. Math. Soc. 119 (1993), 4 1335-1345. (1993) MR1165056DOI10.1090/S0002-9939-1993-1165056-6
  9. Levy R., Matveev M., Some examples of monotonically Lindelöf and not monotonically Lindelöf spaces, Topology Appl. 154 (2007), 2333-2343. (2007) MR2328016
  10. Matveev M., A monotonically Lindelöf space need not be monotonically normal, preprint, 1994. 
  11. Todorčević S., 10.1090/conm/084, Contemporary Mathematics, 84, American Mathematical Society, Providence, Rhode Island, 1989. MR0980949DOI10.1090/conm/084

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