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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.

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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