On the extent of star countable spaces

• Autorzy: Ofelia Alas , Lucia Junqueira , Jan Mill , Vladimir Tkachuk , Richard Wilson
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

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

Kategorie tematyczne

• 54C25: Embedding
• 54C10: Special maps on topological spaces (open, closed, perfect, etc.)
• 22A05: Structure of general topological groups
• 54H11: Topological groups
• 54D25: `` P -minimal'' and `` P -closed'' spaces

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2011
• Tom: 9
• Numer: 3
• Strony: 603-615

Daty

wydano

2011-06-01

online

2011-03-22

Twórcy

autor

Ofelia Alas

• Universidade de São Paulo

autor

Lucia Junqueira

• Universidade de São Paulo

autor

Jan Mill

• VU University Amsterdam

autor

Vladimir Tkachuk

• Universidad Autónoma Metropolitana

autor

Richard Wilson

• Universidad Autónoma Metropolitana

Bibliografia

• [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
• [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
• [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
• [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
• [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
• [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
• [15] Shakhmatov D.B., On pseudocompact spaces with point-countable base, Dokl. Akad. Nauk SSSR, 1984, 30(3), 747–751
• [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
• [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

Identyfikatory

DOI

10.2478/s11533-011-0018-y