On the extent of star countable spaces

Ofelia Alas; Lucia Junqueira; Jan Mill; Vladimir Tkachuk; Richard Wilson

Open Mathematics (2011)

  • Volume: 9, Issue: 3, page 603-615
  • ISSN: 2391-5455

Abstract

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For a topological property P, we say that a space X is star Pif for every open cover Uof the space X there exists Y ⊂ X such that St(Y,U) = X and Y has P. We consider star countable and star Lindelöf spaces establishing, among other things, that there exists first countable pseudocompact spaces which are not star Lindelöf. We also describe some classes of spaces in which star countability is equivalent to countable extent and show that a star countable space with a dense σ-compact subspace can have arbitrary extent. It is proved that for any ω 1-monolithic compact space X, if C p(X)is star countable then it is Lindelöf.

How to cite

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Ofelia Alas, et al. "On the extent of star countable spaces." Open Mathematics 9.3 (2011): 603-615. <http://eudml.org/doc/269470>.

@article{OfeliaAlas2011,
abstract = {For a topological property P, we say that a space X is star Pif for every open cover Uof the space X there exists Y ⊂ X such that St(Y,U) = X and Y has P. We consider star countable and star Lindelöf spaces establishing, among other things, that there exists first countable pseudocompact spaces which are not star Lindelöf. We also describe some classes of spaces in which star countability is equivalent to countable extent and show that a star countable space with a dense σ-compact subspace can have arbitrary extent. It is proved that for any ω 1-monolithic compact space X, if C p(X)is star countable then it is Lindelöf.},
author = {Ofelia Alas, Lucia Junqueira, Jan Mill, Vladimir Tkachuk, Richard Wilson},
journal = {Open Mathematics},
keywords = {Lindelöf property; Extent; Star properties; Star countable spaces; Star Lindelöf spaces; Pseudocompact spaces; Countably compact spaces; Function spaces; κ-monolithic spaces; Products of ordinals; P-spaces; Metalindelöf spaces; Discrete subspaces; Open expansions; extent; star properties; star countable spaces; star Lindelöf spaces; pseudocompact spaces; countably compact spaces; function spaces; -monolithic spaces; products of ordinals; -spaces; metalindelöf spaces; discrete subspaces; open expansions},
language = {eng},
number = {3},
pages = {603-615},
title = {On the extent of star countable spaces},
url = {http://eudml.org/doc/269470},
volume = {9},
year = {2011},
}

TY - JOUR
AU - Ofelia Alas
AU - Lucia Junqueira
AU - Jan Mill
AU - Vladimir Tkachuk
AU - Richard Wilson
TI - On the extent of star countable spaces
JO - Open Mathematics
PY - 2011
VL - 9
IS - 3
SP - 603
EP - 615
AB - For a topological property P, we say that a space X is star Pif for every open cover Uof the space X there exists Y ⊂ X such that St(Y,U) = X and Y has P. We consider star countable and star Lindelöf spaces establishing, among other things, that there exists first countable pseudocompact spaces which are not star Lindelöf. We also describe some classes of spaces in which star countability is equivalent to countable extent and show that a star countable space with a dense σ-compact subspace can have arbitrary extent. It is proved that for any ω 1-monolithic compact space X, if C p(X)is star countable then it is Lindelöf.
LA - eng
KW - Lindelöf property; Extent; Star properties; Star countable spaces; Star Lindelöf spaces; Pseudocompact spaces; Countably compact spaces; Function spaces; κ-monolithic spaces; Products of ordinals; P-spaces; Metalindelöf spaces; Discrete subspaces; Open expansions; extent; star properties; star countable spaces; star Lindelöf spaces; pseudocompact spaces; countably compact spaces; function spaces; -monolithic spaces; products of ordinals; -spaces; metalindelöf spaces; discrete subspaces; open expansions
UR - http://eudml.org/doc/269470
ER -

References

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  1. [1] Alas O.T., Junqueira L.R., Wilson R.G., Countability and star covering properties, Topology Appl., 2011, 158(4), 620–626 http://dx.doi.org/10.1016/j.topol.2010.12.012 Zbl1226.54023
  2. [2] Arkhangel’skii A.V., Structure and classification of topological spaces and cardinal invariants, Uspekhi Mat. Nauk, 1978, 33(6), 29–84 (in Russian) 
  3. [3] Arkhangel’skii A.V., Topological Function Spaces, Math. Appl. (Soviet Ser.), 78, Kluwer, Dordrecht, 1992 
  4. [4] Bonanzinga M., Matveev M.V., Centered-Lindelöfness versus star-Lindelöfness, Comment. Math. Univ. Carolin., 2000, 41(1), 111–122 Zbl1037.54502
  5. [5] van Douwen E.K., Reed G.M., Roscoe A.W., Tree I.J., Star covering properties, Topology Appl., 1991, 39(1), 71–103 http://dx.doi.org/10.1016/0166-8641(91)90077-Y Zbl0743.54007
  6. [6] Dow A., Junnila H., Pelant J., Weak covering properties of weak topologies, Proc. Lond. Math. Soc., 1997, 75(2), 349–368 http://dx.doi.org/10.1112/S0024611597000385 Zbl0886.54014
  7. [7] Engelking R., General Topology, Monografie Matematyczne, 60, PWN, Warszawa, 1977 
  8. [8] Ikenaga S., A class which contains Lindelöf spaces, separable spaces and countably compact spaces, Memoirs of Numazu College of Technology, 1983, 18, 105–108 
  9. [9] Ikenaga S., Somepropertiesofω-n-starspaces, Research Reports of Nara Technical College, 1987, 23, 53–57 
  10. [10] Ikenaga S., Topological concepts between ‘Lindelöf’ and ‘pseudo-Lindelöf’, Research Reports of Nara Technical College, 1990, 26, 103–108 
  11. [11] Ikenaga S., Tani T., On a topological concept between countable compactness and pseudocompactness, Memoirs of Numazu College of Technology, 1980, 15, 139–142 
  12. [12] Matveev M.V., A survey on star covering properties, Topology Atlas, 1998, preprint #330, available at http://at.yorku.ca/v/a/a/a/19.htm 
  13. [13] Matveev M.V., How weak is weak extent?, Topology Appl., 2002, 119(2), 229–232 http://dx.doi.org/10.1016/S0166-8641(01)00061-X Zbl0986.54003
  14. [14] van Mill J., Tkachuk V.V., Wilson R.G., Classes defined by stars and neighbourhood assignments, Topology Appl., 2007, 154(10), 2127–2134 http://dx.doi.org/10.1016/j.topol.2006.03.029 Zbl1131.54022
  15. [15] Shakhmatov D.B., On pseudocompact spaces with point-countable base, Dokl. Akad. Nauk SSSR, 1984, 30(3), 747–751 Zbl0598.54010
  16. [16] Tkachuk V.V., Monolithic spaces and D-spaces revisited, Topology Appl., 2009, 156(4), 840–846 http://dx.doi.org/10.1016/j.topol.2008.11.001 Zbl1165.54009
  17. [17] Williams N.H., Combinatorial Set Theory, Stud. Logic Found. Math., 91, North-Holland, Amsterdam-New York-Oxford, 1977 http://dx.doi.org/10.1016/S0049-237X(08)70663-3 

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