Mary Ellen Rudin's Contributions to the Theory of Nonmetrizable Manifolds

Hereditarily normal manifolds of dimension > 1 may all be metrizable

“…Mary Ellen Rudin proved that MA + ∼CH implies every perfectly normal manifold is metrizable [17]. Hereditary normality (T 5 ) is a natural weakening of perfect normality; Peter Nyikos noticed that, although the Long Line and Long Ray are hereditarily normal non-metrizable manifolds, and indeed the only 1-dimensional non-metrizable connected manifolds [12], it is difficult to find examples of dimension > 1 (although one can do so with ♦ [17] or CH [18]). He therefore raised the problem of whether it was consistent that there weren't any [11], [12].…”

“…Hereditary normality (T 5 ) is a natural weakening of perfect normality; Peter Nyikos noticed that, although the Long Line and Long Ray are hereditarily normal non-metrizable manifolds, and indeed the only 1-dimensional non-metrizable connected manifolds [12], it is difficult to find examples of dimension > 1 (although one can do so with ♦ [17] or CH [18]). He therefore raised the problem of whether it was consistent that there weren't any [11], [12]. In a series of papers [13,14,15,16] he was finally able to prove this from the consistency of a supercompact cardinal, if he also assumed that the manifolds were hereditarily collectionwise Hausdorff.…”

Jordan and Schoenflies in non-metrical analysis situs

“…By the aforementioned (Samelson's Lemma D) it follows that M is compact, hence W is also compact (because W is closed in M). Now since W ext is non-compact (else its interior would be metric), Lemma 30 Irreducible would perhaps be a more neutral terminology. 31 Compare [Schoenflies, 1906] "versus" [Osgood, 1903]: a thoroughgoing account is to be found in Siebenmann [39, §4, Historical notes.…”

Manifolds at and beyond the limit of metrisability

“…A space is ω -bounded if every countable subset has compact closure, and a long pipe is a manifold-with-boundary which is the union of an increasing ω 1 -sequence of open subsets each of which is homeomorphic to S 1 × [0, 1). The paper [16] gives an excellent introduction to the theory of non-metrisable manifolds, while [17] provides a more recent view.…”