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Two nonmetrizable manifolds. (English) Zbl 0708.54012
The author showed in 1979 that the existence of separable perfectly normal non-metrizable manifolds is undecidable in ZFC [Houston J. Math. 5, 249-252 (1979; Zbl 0418.03036)]. Here it is shown that a separable normal nonmetrizable manifold exists in ZFC. The other nonmetrizable manifold has been constructed using \(\diamondsuit^+\). It is normal but not collectionwise normal. As was shown by F. D. Tall [Handbook of set-theoretic topology, 685-732 (1984; Zbl 0552.54011)], adding weakly compact many random reals, normal implies collectionwise normal in manifolds.
Reviewer: A.Szymański

54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
54A35 Consistency and independence results in general topology
03E45 Inner models, including constructibility, ordinal definability, and core models
Full Text: DOI
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