Editing Surface (topology) (section)

== Surfaces with boundary ==
[[Compact manifold|Compact]] surfaces, possibly with boundary, are simply closed surfaces with a finite number of holes (open discs that have been removed).  Thus, a connected compact surface is classified by the number of boundary components and the genus of the corresponding closed surface – equivalently, by the number of boundary components, the orientability, and Euler characteristic. The genus of a compact surface is defined as the genus of the corresponding closed surface.<ref>{{citation
 | last1 = Altınok | first1 = Selma
 | last2 = Bhupal | first2 = Mohan
 | contribution = Minimal page-genus of Milnor open books on links of rational surface singularities
 | doi = 10.1090/conm/475/09272
 | isbn = 978-0-8218-4717-6
 | mr = 2454357
 | pages = 1–10
 | publisher = Amer. Math. Soc., Providence, RI
 | series = Contemp. Math.
 | title = Singularities II
 | volume = 475
 | year = 2008}}; see [https://books.google.com/books?hl=en&lr=&id=uc4bCAAAQBAJ&pg=PA2 p.2]: "Recall that the genus of a compact surface S with boundary is defined to be the genus of the associated closed surface obtained ... by sewing a disc onto each boundary circle"</ref>

This classification follows almost immediately from the classification of closed surfaces: removing an open disc from a closed surface yields a compact surface with a circle for boundary component, and removing ''k'' open discs yields a compact surface with ''k'' disjoint circles for boundary components.  The precise locations of the holes are irrelevant, because the [[homeomorphism group]] acts [[transitive action|''k''-transitively]] on any connected manifold of dimension at least 2.

Conversely, the boundary of a compact surface is a closed 1-manifold, and is therefore the disjoint union of a finite number of circles; filling these circles with disks (formally, taking the [[Cone (topology)|cone]]) yields a closed surface.

The unique compact orientable surface of genus ''g'' and with ''k'' boundary components is often denoted <math>\Sigma_{g,k},</math> for example in the study of the [[mapping class group]].