Bagpipe theorem

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In mathematics, the bagpipe theorem of Peter Nyikos 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".^[1]

Statement

A space is called ω-bounded if the closure of every countable set is compact. Compactness implies ω-boundedness, but in general the converse is not true. For example, the first uncountable ordinal $ω_{1}$ with the order topology and the long line are both ω-bounded but not compact. For metric spaces, however, the two concepts are equivalent.

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 (defined below).

A space P is called a long pipe if there exist subspaces ${U_{α} : α < ω_{1}}$ each of which is homeomorphic to $S^{1} \times ℝ$ such that for $n < m$ , $\overline{U_{n}} \subseteq U_{m}$ and the boundary of $U_{n}$ in $U_{m}$ is homeomorphic to $S^{1}$ . The simplest example of a long 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 $ω_{1}$ copies of the half-open interval $[0, 1)$ , pasted together with the lexicographic ordering. Another long pipe is obtained by removing a single point from the "long plane" $L \times L$ , where $L$ is the long line. There are in fact $2^{ℵ_{1}}$ different isomorphism classes of long pipes.