| 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. (English) |
| 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 |
| . |
