Bagpipe theorem

In mathematics, the bagpipe theorem of Peter Nyikos describes the structure of the connected (but possibly non-paracompact) ω-bounded surfaces by showing that they are "bagpipes": the connected sum of a compact "bag" with several "long pipes".^[1]

Statement

A space is called ω-bounded if the closure of every countable set is compact. Compactness implies ω-boundedness, but in general the converse is not true. For example, the first uncountable ordinal ω 1 {\displaystyle \omega _{1}} with the order topology and the long line are both ω-bounded but not compact. For metric spaces, however, the two concepts are equivalent.

The bagpipe theorem states that every ω-bounded connected surface is the connected sum of a compact connected surface and a finite number of long pipes (defined below).

A space P is called a long pipe if there exist subspaces { U α : α < ω 1 } {\displaystyle \{U_{\alpha }:\alpha <\omega _{1}\}} each of which is homeomorphic to S 1 × R {\displaystyle S^{1}\times \mathbb {R} } such that for n < m {\displaystyle n<m} , U n ¯ ⊆ U m {\displaystyle {\overline {U_{n}}}\subseteq U_{m}} and the boundary of U n {\displaystyle U_{n}} in U m {\displaystyle U_{m}} is homeomorphic to S 1 {\displaystyle S^{1}} . The simplest example of a long pipe is the product S 1 × L + {\displaystyle S^{1}\times L^{+}} of the circle S 1 {\displaystyle S^{1}} and the long closed ray L + {\displaystyle L^{+}} , which is an increasing union of ω 1 {\displaystyle \omega _{1}} copies of the half-open interval [ 0 , 1 ) {\displaystyle [0,1)} , pasted together with the lexicographic ordering. Another long pipe is obtained by removing a single point from the "long plane" L × L {\displaystyle L\times L} , where L {\displaystyle L} is the long line. There are in fact 2 ℵ 1 {\displaystyle 2^{\aleph _{1}}} different isomorphism classes of long pipes.

The bagpipe theorem does not describe all surfaces since there are many examples of surfaces that are not ω-bounded, such as the Prüfer manifold.

References