Editing Surface (topology) (section)

== Non-compact surfaces ==
Non-compact surfaces are more difficult to classify.  As a simple example, a non-compact surface can be obtained by puncturing (removing a finite set of points from) a closed manifold.  On the other hand, any open subset of a compact surface is itself a non-compact surface; consider, for example, the complement of a [[Cantor set]] in the sphere, otherwise known as the [[Cantor tree surface]].  However, not every non-compact surface is a subset of a compact surface; two canonical counterexamples are the [[Jacob's ladder (manifold)|Jacob's ladder]] and the [[Loch Ness monster surface|Loch Ness monster]], which are non-compact surfaces with infinite genus.

A non-compact surface ''M'' has a non-empty [[End (topology)|space of ends]] ''E''(''M''), which informally speaking describes the ways that the surface "goes off to infinity".  The space ''E''(''M'') is always topologically equivalent to a closed subspace of the [[Cantor set]].  ''M'' may have a finite or countably infinite number N<sub>h</sub> of handles, as well as a finite or countably infinite number ''N''<sub>''p''</sub> of [[projective plane]]s. If both ''N''<sub>''h''</sub> and ''N''<sub>''p''</sub> are finite, then these two numbers, and the topological type of space of ends, classify the surface ''M'' up to topological equivalence.  If either or both of ''N''<sub>''h''</sub> and ''N''<sub>''p''</sub> is infinite, then the topological type of M depends not only on these two numbers but also on how the infinite one(s) approach the space of ends. In general the topological type of M is determined by the four subspaces of ''E''(''M'') that are limit points of infinitely many handles and infinitely many projective planes, limit points of only handles, limit points of only projective planes, and limit points of neither.<ref>{{cite journal |last1=Richards |first1=Ian |date=1963 |title=On the classification of noncompact surfaces |journal=Trans. Amer. Math. Soc. |volume=106 |issue=2 |pages=259–269 |doi=10.2307/1993768 |jstor=1993768 |doi-access=free }}</ref>