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 <math>\{U_\alpha: \alpha < \omega_1 \} </math> each of which is homeomorphic to <math>S^1 \times \mathbb{R} </math> such that for <math>n<m</math> we have <math> \overline{U_n} \subseteq U_m</math> and the boundary of <math>U_n</math> in <math>U_m</math> is homeomorphic to <math>S^1</math>. The simplest example of a pipe is the product <math>S^1 \times L^+</math> of the circle <math>S^1</math> and the long closed ray <math>L^+</math>, which is an increasing union of <math>\omega_1</math> copies of the half-open interval <math>[0,1)</math>, pasted together with the lexicographic ordering. Here, <math>\omega_1</math> 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" <math>L \times L</math> where <math>L</math> is the long line, formed by gluing together two copies of <math>L^+</math> at their endpoints to get a space which is "long at both ends". There are in fact <math>2^{\aleph_1}</math> 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.
