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

Abstract

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Given a topological property (or a class) 𝒫 , the class 𝒫 * dual to 𝒫 (with respect to neighbourhood assignments) consists of spaces X such that for any neighbourhood assignment { O x : x ∈ X } there is Y ⊂ X with Y ∈ 𝒫 and ⋃ { O x : x ∈ Y } = X . The spaces from 𝒫 * are called dually 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.

How to cite

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Buzyakova, 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

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  1. 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
  2. Alas O.T., Tkachuk V.V., Wilson R.G., Covering properties and neighbourhood assignments, Topology Proc. 30 1 (2006), 25-37. (2006) MR2280656
  3. 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
  4. 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
  5. Engelking R., General Topology, PWN, Warszawa, 1977. Zbl0684.54001MR0500780
  6. 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
  7. 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
  8. Ostaszewski A., On countably compact, perfectly normal spaces, J. London Math. Soc. 14 2 (1976), 505-516. (1976) Zbl0348.54014MR0438292
  9. 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
  10. Todorcevic S., Partition Problems in Topology, Contemporary Math. 84, Amer. Math. Soc., Providence, RI, 1989. Zbl0659.54001MR0980949

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