Bagpipe theorem

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 $$S^1 \times \mathbb{R} $$ 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 $$S^1 \times L^+$$ 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" $$L \times L$$ 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.