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2 Because of notes by dear Rivin, I wrote some details to precision the question.

Maybe this question be very simple, but I don't know why it is hard for me. Thanks for any guide and help.

We say a surface $S$ (2-dimensional metric(compact) Riemannian surface) is good(denote good (denote by GS), $GS$), if every $2n$, $n\geq1$, points on surface can be separate by some geodesic to two distinct subsets $V_1$ and $V_2$, where $|V_1|=|V_2|=n$. Also, if a surface $S$ is not good, we say it is bad an denote it by $BS$.

For example, it is not difficult to show that plane is a $GS$. Also, a sphere is $GS$.

1) Do we have some $BS$ examples(class of examples)?

2) Can we characterize the $GS$ and $BS$ surfaces?

I can't find any $BS$ examples and also I can't prove that they are $GS$.

For example, is Klein Bottle $GS$ or $BS$?