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A quest for nice kernels of neighbourhood assignments. (English) Zbl 1199.54141
Summary: 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 $$\{O_x\:x\in X\}$$, there is $$Y\subset X$$ with $$Y\in \mathcal P$$ and $$\bigcup \{O_x\:x\in Y\}=X$$. 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.

##### MSC:
 54D20 Noncompact covering properties (paracompact, Lindelöf, etc.) 54H11 Topological groups (topological aspects) 54C10 Special maps on topological spaces (open, closed, perfect, etc.) 22A05 Structure of general topological groups
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