## R.Z. Buzyakova, V.V. Tkachuk, R.G. Wilson

*A quest for nice kernels of neighbourhood assignments *

Comment.Math.Univ.Carolin. 48,4 (2007) 689-697. **Abstract:**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 {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.

**Keywords:** neighbourhood assignment, duality, weak duality, Lindel\"of space, weakly Lindel\"of space

**AMS Subject Classification:** Primary 54H11, 54C10, 22A05, 54D06; Secondary 54D25, 54C25