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Classes defined by stars and neighbourhood assignments. (English) Zbl 1131.54022
A neighbourhood assignment in a space \(X\) is a family \(\{O_x:\;x\in X\}\) such that \(x\in O_x\in \tau (X)\) for any \(x\in X\). Given a topological property \(\mathcal P\), 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 \(Y\subseteq X\) with \(Y\in \mathcal P\) and \(\bigcup\{O_x:\;x\in Y\}=X\). The authors show that compactness, \(\kappa\)-compactness, pseudocompactness and linear Lindelöfness are self-dual in this sense. Given a family \(\mathcal U\) of subsets of \(X\), the star, \(St(A,{\mathcal U})\), of the set \(A\) with respect to \(\mathcal U\) is the set \(\bigcup \{U\in{\mathcal U}:A\cap U\neq\emptyset\}\). \(X\) is said to be star-\(\mathcal P\) (or star determined by \(\mathcal P\)), if for any open cover \(\mathcal U\) of the space \(X\) there is a subspace \(Y\subseteq X\), with \(St(Y,{\mathcal U})=X\) and \(Y\in \mathcal P\). The authors show that the classes of pseudocompact spaces, the spaces star determined by countably compact spaces, star compact spaces, and countably compact spaces are all distinct. Among other things, an example is given of (1) A pseudocompact first countable space which is not star determined by countably compact spaces. (2) A space \(X\) which is star determined by metrizable compact spaces but not compact countable spaces. Several problems are left unsolved, such as: Must every star compact topological group be countably compact?

MSC:
54H11 Topological groups (topological aspects)
54C10 Special maps on topological spaces (open, closed, perfect, etc.)
22A05 Structure of general topological groups
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54D25 “\(P\)-minimal” and “\(P\)-closed” spaces
54C25 Embedding
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