= Boundary component of a surface
{title2=$b=\#\pi_0(\partial M)$}
= Boundary components of a surface
{synonym}
A boundary component of a <surface with boundary> is a <connected component> of its <manifold boundary>. For a <compact> <surface with boundary>, these are finitely many <circles>. Their number $b$ enters the <classification of compact surfaces>: $\chi=2-2g-b$ in the <orientable> case and $\chi=2-k-b$ in the <nonorientable surface> case, where $k$ is the <nonorientable genus>.
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