Boundary component of a surface ($b=\#{\pi }_0(\partial M)$)

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.