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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

##### MSC:
 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
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##### References:
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