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Type I Manifolds and the Bagpipe Theorem

  • David Gauld
Chapter

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 \(\aleph _1\) many open Lindelöf subspaces rather than \(\aleph _0\) many (hence, in fact, a single one). He also presented a condition, called \({\omega }\)-boundedness, which is equivalent to compactness in a metric space but not in a general topological space. A manifold is \({\omega }\)-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 \({\omega }\)-bounded surfaces. We present a proof of Nyikos’s Bagpipe Theorem. We also show that there are \(2^{\aleph _1}\) many \({\omega }\)-bounded, connected surfaces: contrast this with the compact, connected surfaces of which there are only \(\aleph _0\) many.

Keywords

Compact Manifold Boundary Component Topological Type Open Unit Disc Connected Surface 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media Singapore 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of AucklandAucklandNew Zealand

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