Bešlagić, Amer; Rudin, Mary Ellen Set-theoretic constructions of nonshrinking open covers. (English) Zbl 0574.54020 Topology Appl. 20, 167-177 (1985). The authors give two interesting examples of topological spaces. (1) Let \(\kappa\) be an infinite regular cardinal. Then \(\diamond^{++}\) implies the existence of a Hausdorff, strongly zero-dimensional, collectionwise normal, \(\kappa\)-ultraparacompact, P-space with the following property: Every increasing open cover has a clopen shrinking, but there is an open cover which has no closed shrinking. (2) Let \(\kappa\) be an uncountable regular cardinal. Then \(\Delta\) implies the existence of a Hausdorff, collectionwise normal, countably ultraparacompact P-space, which has a strictly increasing open cover having no shrinking, each member of which is the union of at most \(\kappa\) closed sets. Reviewer: R.Z.Domiaty Cited in 1 ReviewCited in 7 Documents MSC: 54D20 Noncompact covering properties (paracompact, Lindelöf, etc.) 54A35 Consistency and independence results in general topology 03E45 Inner models, including constructibility, ordinal definability, and core models 54G10 \(P\)-spaces Keywords:shrinkable normal P-space; \(V=L\); Hausdorff, strongly zero-dimensional, collectionwise normal, \(\kappa \) -ultraparacompact, P-space; clopen shrinking; Hausdorff, collectionwise normal, countably ultraparacompact P-space; strictly increasing open cover PDFBibTeX XMLCite \textit{A. Bešlagić} and \textit{M. E. Rudin}, Topology Appl. 20, 167--177 (1985; Zbl 0574.54020) Full Text: DOI References: [1] Engelking, R., General Topology (1977), Polish Scientific Publishers: Polish Scientific Publishers Warszawa [2] Smith, J. C., Applications of shrinkable covers, Proc. Amer. Math. Soc., 73, 379-387 (1979) · Zbl 0367.54007 [3] Chiba, K., On the weak \(B\)-property and ∑-products, Math. 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