Bagpipe theorem explained

In mathematics, the bagpipe theorem of 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".

Statement

A space is called ω-bounded if the closure of every countable set is compact. For example, the long line and the closed long ray are ω-bounded but not compact. When restricted to a metric space ω-boundedness is equivalent to compactness.

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.

A space P is called a long pipe if there exist subspaces

\{U _{\alpha:} \alpha < \omega _{1} \} each of which is homeomorphic to S ^{1} x R such that for n<m we have \overline{U _{n}} \subseteq U _{m} and the boundary of U _{n} in U _{m} is homeomorphic to S ^{1} . The simplest example of a pipe is the product S ^{1} x L ^{+} of the circle S ^{1} and the long closed ray L ^{+} , which is an increasing union of \omega _{1} copies of the half-open interval [ 0 , 1 ) , pasted together with the lexicographic ordering. Here, \omega _{1} denotes the first uncountable ordinal number, which is the set of all countable ordinals. Another (non-isomorphic) example is given by removing a single point from the "long plane" L x L where L is the long line, formed by gluing together two copies of L ^{+} at their endpoints to get a space which is "long at both ends". There are in fact 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.