# 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

## Access Full Article

top## Abstract

top## How to cite

topOfelia 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

top- [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] Arkhangel’skii A.V., Structure and classification of topological spaces and cardinal invariants, Uspekhi Mat. Nauk, 1978, 33(6), 29–84 (in Russian)
- [3] Arkhangel’skii A.V., Topological Function Spaces, Math. Appl. (Soviet Ser.), 78, Kluwer, Dordrecht, 1992
- [4] Bonanzinga M., Matveev M.V., Centered-Lindelöfness versus star-Lindelöfness, Comment. Math. Univ. Carolin., 2000, 41(1), 111–122 Zbl1037.54502
- [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] 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] Engelking R., General Topology, Monografie Matematyczne, 60, PWN, Warszawa, 1977
- [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] Ikenaga S., Somepropertiesofω-n-starspaces, Research Reports of Nara Technical College, 1987, 23, 53–57
- [10] Ikenaga S., Topological concepts between ‘Lindelöf’ and ‘pseudo-Lindelöf’, Research Reports of Nara Technical College, 1990, 26, 103–108
- [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] 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] 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] 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] Shakhmatov D.B., On pseudocompact spaces with point-countable base, Dokl. Akad. Nauk SSSR, 1984, 30(3), 747–751 Zbl0598.54010
- [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] 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

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.