Bagpipe theorem

In mathematics, the bagpipe theorem of Peter Nyikos (1984) 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.

References

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