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A glance at spaces with closure-preserving local bases. (English summary) Topology Appl. 157 (2010), no. 3, 548–558. The authors define a space X to be (weakly) Japanese if there is a closure-preserving local (quasi-) base at each point x of X. In the paper under review many interesting results are established about Japanese and weakly Japanese spaces. They prove, for example, that every GO space is Japanese, and that the property of being Japanese, or weakly Japanese, is preserved by σ-products. They also show that if M is a product of uncountably many spaces each of which has at least two elements, then no Gδ-dense subspace of M is weakly Japanese at any of its elements, in the sense that no point has a closure-preserving local quasi-base. The authors discuss compact spaces, and show that while scattered Corson compact spaces are hereditarily Japanese, even an Eberlein compact (non-scattered) space need not be weakly Japanese. In the final theorem in the article, the authors show that assuming ♦ there is a crowded hereditarily separable compact space having no nontrivial convergent sequences and which fails to be weakly Japanese at each of its points. The article closes with a list of open questions, the first of which asks if every weakly Japanese space is Japanese

Year: 2013

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