Non-metrisable Manifolds pp 49-62 | Cite as

# Type I Manifolds and the Bagpipe Theorem

## 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## References

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