i have a question. I think that is not difficult, but i can't find a solution.7 I want to show the following:
Cutting a sphere along a curve always results in two discs.
Therefore i want to use the classification theorem and the Euler characteristic. Here my own ideas: Notice that we have a sphere, thus every curve seperates the surface and $\chi(F)=2$ (in addition these two facts are equivalent). But then, cutting along a curve must give: $$2=\chi(F_1)+\chi(F_2)$$ But $\chi(F_i)\leq2$ because the Euler-characteristic is always lower of equal to $2$. But then we have two possbilities: $2=2+0$ or $2=1+1$. The first can not be true because every curve seperates the sphere. Thus it must be the second one. But then we may conclude that we have two discs.
My Problem: I don't know if i can make such a argumentation. I think i miss something, true?!
Thank you for help, hints and solutions.