Are Perfectly Normal Manifolds Metrisable?
A big challenge for Set Theory for many years was whether the Continuum Hypothesis CH or its negation \(\lnot \) CH followed from the usual axioms of Set Theory, ZFC. In the 1930s Gödel showed that CH was at least consistent with ZFC but then in the 1960s Cohen showed that \(\lnot \) CH is also consistent with ZFC: so CH is independent of ZFC. Then in the 1970s the answer to a long-standing question in the topology of manifolds, whether every perfectly normal manifold is metrisable, was found to be independent of ZFC too. In this chapter we exhibit (essentially) the perfectly normal, non-metrisable manifold which Rudin and Zenor constructed using CH. We also present Rudin’s proof that under MA \(+\lnot \) CH every perfectly normal manifold is metrisable.
KeywordsOpen Cover Accumulation Point Continuum Hypothesis Usual Topology Countable Dense Subset
- 2.Gauld, D.: A strongly hereditarily separable non-metrisable manifold. Top. Appl. 51, 221–228 (1993)Google Scholar
- 3.Kozslowski, G., Zenor, P.: A differentiable, perfectly normal nonmetrizable manifold. Topol. Proc. 4, 453–461 (1979)Google Scholar
- 4.Roitman, J.: Basic S and L. In: Kunen, K., Vaughan, J.E. (eds.) Handbook of Set-Theoretic Topology, pp. 295–326. North-Holland, Amsterdam (1984)Google Scholar
- 5.Rudin, M.E.: The undecidability of the existence of a perfectly normal nonmetrizable manifold. Houst. J. Math. 5, 249–252 (1979)Google Scholar
- 6.Rudin, M.E., Zenor, P.: A perfectly normal nonmetrizable manifold. Houst. J. Math. 2, 129–134 (1976)Google Scholar
- 7.Szentmiklóssy, Z.: S-spaces and L-spaces under Martin’s axiom. Coll. Math. Soc. János Bolyai 23, 1139–1145 (1978)Google Scholar
- 8.Wilder, R.L.: Topology of Manifolds. Amer. Math. Soc. vol. 32. Colloquium Publications, New York (1949)Google Scholar