# Article

 Title: A quest for nice kernels of neighbourhood assignments  (English) Author: Buzyakova, R. Z. Author: Tkachuk, V. V. Author: Wilson, R. G. Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 48 Issue: 4 Year: 2007 Pages: 689-697 . Category: math . Summary: Given a topological property (or a class) $\Cal P$, the class $\Cal P^*$ dual to $\Cal P$ (with respect to neighbourhood assignments) consists of spaces $X$ such that for any neighbourhood assignment $\{O_x:x\in X\}$ there is $Y\subset X$ with $Y\in \Cal P$ and $\bigcup\{O_x:x\in Y\}=X$. The spaces from $\Cal P^*$ are called {\it dually $\Cal P$\/}. We continue the study of this duality which constitutes a development of an idea of E. van Douwen used to define $D$-spaces. We prove a number of results on duals of some general classes of spaces establishing, in particular, that any generalized ordered space of countable extent is dually discrete. Keyword: neighbourhood assignment Keyword: duality Keyword: weak duality Keyword: Lindelöf space Keyword: weakly Lindelöf space MSC: 22A05 MSC: 54C10 MSC: 54C25 MSC: 54D06 MSC: 54D20 MSC: 54D25 MSC: 54H11 idZBL: Zbl 1199.54141 idMR: MR2375169 . Date available: 2009-05-05T17:05:43Z Last updated: 2012-05-01 Stable URL: http://hdl.handle.net/10338.dmlcz/119691 . Reference: [AB] Arhangel'skii A.V., Buzyakova R.Z.: Convergence in compacta and linear Lindelöfness.Comment. Math. Univ. Carolin. 39 1 (1998), 159-166. Zbl 0937.54022, MR 1623006 Reference: [ATW] Alas O.T., Tkachuk V.V., Wilson R.G.: Covering properties and neighbourhood assignments.Topology Proc. 30 1 (2006), 25-37. MR 2280656 Reference: [DTTW] Dow A., Tkachenko M.G., Tkachuk V.V., Wilson R.G.: Topologies generated by discrete subspaces.Glas. Mat. Ser. III 37(57) (2002), 1 187-210. Zbl 1009.54005, MR 1918105 Reference: [vDL] van Douwen E.K., Lutzer D.J.: A note on paracompactness in generalized ordered spaces.Proc. Amer. Math. Soc. 125 4 (1997), 1237-1245. Zbl 0885.54023, MR 1396999 Reference: [En] Engelking R.: General Topology.PWN, Warszawa, 1977. Zbl 0684.54001, MR 0500780 Reference: [Lu] Lutzer D.J.: Ordered Topological Spaces.Surveys in General Topology, ed. by G.M. Reed, Academic Press, New York, 1980, pp. 247-295. Zbl 0472.54020, MR 0564104 Reference: [vMTW] van Mill J., Tkachuk V.V., Wilson R.G.: Classes defined by stars and neighbourhood assignments.Topology Appl. 154 (2007), 2127-2134. Zbl 1131.54022, MR 2324924 Reference: [Os] Ostaszewski A.: On countably compact, perfectly normal spaces.J. London Math. Soc. 14 2 (1976), 505-516. Zbl 0348.54014, MR 0438292 Reference: [Ro] Roitman J.: Basic $S$ and $L$.Handbook of Set-Theoretic Topology, ed. by K. Kunen and J.E. Vaughan, Elsevier S.P. B.V., Amsterdam, 1984, pp.295-326. Zbl 0594.54001, MR 0776626 Reference: [To] Todorcevic S.: Partition Problems in Topology.Contemporary Math. 84, Amer. Math. Soc., Providence, RI, 1989. Zbl 0659.54001, MR 0980949 .

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