On the extent of star countable spaces. (English) Zbl 1246.54017
For a topological property \(\mathcal P\), a space \(X\) is star \(\mathcal P\) if for every open cover \(\mathcal U\) of \(X\) there exists \(Y \subset X\) such that \(X=\text{St}(Y,\mathcal U) = \bigcup\{U\in\mathcal U: U\cap Y\neq\emptyset\}\), and \(Y\) has property \(\mathcal P\). Continuing the research done by O. T. Alas, L. R. Junqueira and R. G. Wilson in [Topology Appl. 158, No. 4, 620–626 (2011; Zbl 1226.54023)], the authors study star countable and star Lindelöf spaces establishing, among other things, that there exist first countable pseudocompact spaces which are not star Lindelöf. Also, some classes of spaces are described where star countability is equivalent to countable extent, and it is shown that a star countable space with a dense \(\sigma \)-compact subspace can have arbitrary extent. It is proved that for any \(\omega _{1}\)-monolithic compact space \(X\), if \(C _{p }(X)\) is star countable, then it is Lindelöf. The paper concludes with a list of related open problems.
Reviewer: Laszlo Zsilinszky (Pembroke)
MSC:
54D20 | Noncompact covering properties (paracompact, Lindelöf, etc.) |
54A25 | Cardinality properties (cardinal functions and inequalities, discrete subsets) |
54D60 | Realcompactness and realcompactification |
54C35 | Function spaces in general topology |
54H11 | Topological groups (topological aspects) |
54C25 | Embedding |
Keywords:
Lindelöf property; extent; star properties; star countable spaces; star Lindelöf spaces; pseudocompact spaces; countably compact spaces; function spaces; \(\kappa\)-monolithic spaces; products of ordinals; \(\mathcal P\)-spaces; metalindelöf spaces; discrete subspaces; open expansionsCitations:
Zbl 1226.54023References:
[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; · Zbl 1226.54023 |
[2] | Arkhangel’skii A.V., Structure and classification of topological spaces and cardinal invariants, Uspekhi Mat. Nauk, 1978, 33(6), 29-84 (in Russian); · Zbl 0414.54002 |
[3] | Arkhangel’skii A.V., Topological Function Spaces, Math. Appl. (Soviet Ser.), 78, Kluwer, Dordrecht, 1992; · Zbl 0758.46026 |
[4] | Bonanzinga M., Matveev M.V., Centered-Lindelöfness versus star-Lindelöfness, Comment. Math. Univ. Carolin., 2000, 41(1), 111-122; · Zbl 1037.54502 |
[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; · Zbl 0743.54007 |
[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; · Zbl 0886.54014 |
[7] | Engelking R., General Topology, Monografie Matematyczne, 60, PWN, Warszawa, 1977; · Zbl 0373.54002 |
[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; · Zbl 0986.54003 |
[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; · Zbl 1131.54022 |
[15] | Shakhmatov D.B., On pseudocompact spaces with point-countable base, Dokl. Akad. Nauk SSSR, 1984, 30(3), 747-751; · Zbl 0598.54010 |
[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; · Zbl 1165.54009 |
[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; · Zbl 0362.04008 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.