Classification of surfaces

Min uppgift:

Show that a compact connected 2-manifold with a subspace that is homemorphic to a Möbius band is the connected sum of a compact connected manifold with ${\mathrm{\mathbb{P}}}^2\left(\mathrm{ℝ}\right)$.

Min start: Let S be acompact connected 2-manifold (surface) and M be a subspace homeomorphic to the Möbius band.

We know that M is cloed (since a 2 manifold is a Hausdorff space).

… och det är så långt jag har kommit.