Geodesic
A quest for nice kernels of neighbourhood assignments
Commentationes Mathematicae Universitatis Carolinae, Tome 48 (2007) no. 4, pp. 689-697

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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.
Classification : 22A05, 54C10, 54C25, 54D06, 54D20, 54D25, 54H11
Keywords: neighbourhood assignment; duality; weak duality; Lindelöf space; weakly Lindelöf space 
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     title = {A quest for nice kernels of neighbourhood assignments},
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Buzyakova, R. Z.; Tkachuk, V. V.; Wilson, R. G. A quest for nice kernels of neighbourhood assignments. Commentationes Mathematicae Universitatis Carolinae, Tome 48 (2007) no. 4, pp. 689-697. http://geodesic.mathdoc.fr/item/CMUC_2007__48_4_a11/