Bagpipe theorem

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

• Nyikos, Peter (1984), "The theory of nonmetrizable manifolds", Handbook of set-theoretic topology, Amsterdam: North-Holland, pp. 633–684, MR 0776633