Type I Manifolds and the Bagpipe Theorem

David Gauld ()
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David Gauld: University of Auckland, Department of Mathematics

Chapter Chapter 4 in Non-metrisable Manifolds, 2014, pp 49-62 from Springer

Abstract: Abstract In 1984 Nyikos introduced a special class of manifolds which he called Type I. These are manifolds which might just fail to be metrisable in the sense that they are a union of $$\aleph _1$$ ℵ 1 many open Lindelöf subspaces rather than $$\aleph _0$$ ℵ 0 many (hence, in fact, a single one). He also presented a condition, called $${\omega }$$ ω -boundedness, which is equivalent to compactness in a metric space but not in a general topological space. A manifold is $${\omega }$$ ω -bounded if and only if it is of Type I and is countably compact. Nyikos then went on to prove his amazing Bagpipe Theorem which describes the structure of $${\omega }$$ ω -bounded surfaces. We present a proof of Nyikos’s Bagpipe Theorem. We also show that there are $$2^{\aleph _1}$$ 2 ℵ 1 many $${\omega }$$ ω -bounded, connected surfaces: contrast this with the compact, connected surfaces of which there are only $$\aleph _0$$ ℵ 0 many.

Keywords: Compact Manifold; Boundary Component; Topological Type; Open Unit Disc; Connected Surface (search for similar items in EconPapers)
Date: 2014
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-981-287-257-9_4

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DOI: 10.1007/978-981-287-257-9_4

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