Nonorientable genus
= Nonorientable genus
{title2=$k$}
For a <compact> <connected> <nonorientable surface>, its nonorientable genus $k\geq1$ is the number of <real projective planes> in its <connected sum of surfaces> representation after capping all <boundary components of a surface> by <closed discs>. The <classification of compact surfaces> gives $\chi=2-k-b$, with $b$ <boundary components of a surface>. The <real projective plane> has $k=1$ and the <Klein bottle> has $k=2$.