# A quest for nice kernels of neighbourhood assignments

Raushan Z. Buzyakova; Vladimir Vladimirovich Tkachuk; Richard Gordon Wilson

Commentationes Mathematicae Universitatis Carolinae (2007)

- Volume: 48, Issue: 4, page 689-697
- ISSN: 0010-2628

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topBuzyakova, Raushan Z., Tkachuk, Vladimir Vladimirovich, and Wilson, Richard Gordon. "A quest for nice kernels of neighbourhood assignments." Commentationes Mathematicae Universitatis Carolinae 48.4 (2007): 689-697. <http://eudml.org/doc/250232>.

@article{Buzyakova2007,

abstract = {Given a topological property (or a class) $\mathcal \{P\}$, the class $\mathcal \{P\}^*$ dual to $\mathcal \{P\}$ (with respect to neighbourhood assignments) consists of spaces $X$ such that for any neighbourhood assignment $\lbrace O_x:x\in X\rbrace $ there is $Y\subset X$ with $Y\in \mathcal \{P\}$ and $\bigcup \lbrace O_x:x\in Y\rbrace =X$. The spaces from $\mathcal \{P\}^*$ are called dually $\mathcal \{P\}$. 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.},

author = {Buzyakova, Raushan Z., Tkachuk, Vladimir Vladimirovich, Wilson, Richard Gordon},

journal = {Commentationes Mathematicae Universitatis Carolinae},

keywords = {neighbourhood assignment; duality; weak duality; Lindelöf space; weakly Lindelöf space; neighbourhood assignment; duality; weak duality; Lindelöf space; weakly Lindelöf space},

language = {eng},

number = {4},

pages = {689-697},

publisher = {Charles University in Prague, Faculty of Mathematics and Physics},

title = {A quest for nice kernels of neighbourhood assignments},

url = {http://eudml.org/doc/250232},

volume = {48},

year = {2007},

}

TY - JOUR

AU - Buzyakova, Raushan Z.

AU - Tkachuk, Vladimir Vladimirovich

AU - Wilson, Richard Gordon

TI - A quest for nice kernels of neighbourhood assignments

JO - Commentationes Mathematicae Universitatis Carolinae

PY - 2007

PB - Charles University in Prague, Faculty of Mathematics and Physics

VL - 48

IS - 4

SP - 689

EP - 697

AB - Given a topological property (or a class) $\mathcal {P}$, the class $\mathcal {P}^*$ dual to $\mathcal {P}$ (with respect to neighbourhood assignments) consists of spaces $X$ such that for any neighbourhood assignment $\lbrace O_x:x\in X\rbrace $ there is $Y\subset X$ with $Y\in \mathcal {P}$ and $\bigcup \lbrace O_x:x\in Y\rbrace =X$. The spaces from $\mathcal {P}^*$ are called dually $\mathcal {P}$. 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.

LA - eng

KW - neighbourhood assignment; duality; weak duality; Lindelöf space; weakly Lindelöf space; neighbourhood assignment; duality; weak duality; Lindelöf space; weakly Lindelöf space

UR - http://eudml.org/doc/250232

ER -

## References

top- Arhangel'skii A.V., Buzyakova R.Z., Convergence in compacta and linear Lindelöfness, Comment. Math. Univ. Carolin. 39 1 (1998), 159-166. (1998) Zbl0937.54022MR1623006
- Alas O.T., Tkachuk V.V., Wilson R.G., Covering properties and neighbourhood assignments, Topology Proc. 30 1 (2006), 25-37. (2006) MR2280656
- 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. (2002) Zbl1009.54005MR1918105
- van Douwen E.K., Lutzer D.J., A note on paracompactness in generalized ordered spaces, Proc. Amer. Math. Soc. 125 4 (1997), 1237-1245. (1997) Zbl0885.54023MR1396999
- Engelking R., General Topology, PWN, Warszawa, 1977. Zbl0684.54001MR0500780
- Lutzer D.J., Ordered Topological Spaces, Surveys in General Topology, ed. by G.M. Reed, Academic Press, New York, 1980, pp. 247-295. Zbl0472.54020MR0564104
- van Mill J., Tkachuk V.V., Wilson R.G., Classes defined by stars and neighbourhood assignments, Topology Appl. 154 (2007), 2127-2134. (2007) Zbl1131.54022MR2324924
- Ostaszewski A., On countably compact, perfectly normal spaces, J. London Math. Soc. 14 2 (1976), 505-516. (1976) Zbl0348.54014MR0438292
- 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. Zbl0594.54001MR0776626
- Todorcevic S., Partition Problems in Topology, Contemporary Math. 84, Amer. Math. Soc., Providence, RI, 1989. Zbl0659.54001MR0980949

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