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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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