Bagpipe theorem


In mathematics, the bagpipe theorem of describes the structure of the connected 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 long pipe is roughly an increasing union of ω1 copies of the half-open cylinder. There are different isomorphism classes of long pipes. Two examples of long pipes are the product of a circle with a closed long ray, and the "long plane" with an open disk removed.
There are many examples of surfaces that are not ω-bounded, such as the Prüfer manifold.