×

On weaker forms of the chain \((F)\) condition and metacompactness-like covering properties in the product spaces. (English) Zbl 1338.54128

Summary: We introduce the concept of a family of sets generating another family. Then we prove that if \(X\) is a topological space and \(X\) has \(W = \{W(x): x \in X\}\) which is finitely generated by a countable family satisfying \((F)\) which consists of families each Noetherian of \(\omega \)-rank, then \(X\) is metaLindelöf as well as a countable product of them. We also prove that if \(W\) satisfies \(\omega \)-rank \((F)\) and, for every \(x \in X, W(x)\) is of the form \(W _{0}(x) \cup W _{1}(x)\), where \(W _{0}(x)\) is Noetherian and \(W _{1}(x)\) consists of neighbourhoods of \(x\), then \(X\) is metacompact.

MSC:

54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
03E02 Partition relations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Collins P.J., Reed G.M., Roscoe A.W., Rudin M.E., A lattice of conditions on topological spaces, Proc. Amer. Math. Soc., 1985, 94(3), 487-496 http://dx.doi.org/10.1090/S0002-9939-1985-0787900-X; · Zbl 0562.54043
[2] Erdős P., Hajnal A., Máté A., Rado R., Combinatorial Set Theory: Partition Relations for Cardinals, Stud. Logic Found. Math., 106, North-Holland, Amsterdam-New York-Oxford, 1984; · Zbl 0573.03019
[3] Gartside P.M., Moody P.J., Well-ordered (F) spaces, Topology Proc., 1992, 17, 111-130; · Zbl 0797.54038
[4] Gruenhage G., Nyikos P., Spaces with bases of countable rank, General Topology and Appl., 1978, 8(3), 233-257 http://dx.doi.org/10.1016/0016-660X(78)90004-1; · Zbl 0412.54034
[5] Hajnal A., Hamburger P., Set Theory, London Math. Soc. Stud. Texts, 48, Cambridge University Press, Cambridge, 1999 http://dx.doi.org/10.1017/CBO9780511623561;
[6] Moody P.J., Reed G.M., Roscoe A.W., Collins P.J., A lattice of conditions on topological spaces. II, Fund. Math., 1991, 138(2), 69-81; · Zbl 0745.54008
[7] Nagata J., On dimension and metrization, In: General Topology and its Relations to Modern Analysis and Algebra, Prague, 1961, Academic Press, New York, 1962, 282-285;
[8] Nyikos P.J., Some surprising base properties in topology. II, In: Set-Theoretic Topology, Athens, 1975-1976, Academic Press, New York-London, 1977, 277-305; · Zbl 0337.54014
[9] Vural Ç., Some weaker forms of the chain (F) condition for metacompactness, J. Aust. Math. Soc., 2008, 84(2), 283-288 http://dx.doi.org/10.1017/S1446788708000037; · Zbl 1151.54020
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.