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Proof:

Ok, here are some of my thoughts about the proof.

I believe that one would need to use the following theorem(while it is possible that it can be done in other ways as well).

Thm.

*Any compact surface is either homeomorphic to a sphere, or to a connected sum of tori, or to a connected sum of projective planes.*

The part that throws me off is "Suppose that the sides are identified in pairs in accordance with

**any symbol whatsoever**". Do they mean that any two sides of the n-gon can be identified with one another or?

Proof: If it is a 2-gon, then there are really only two ways one could identify its sides in pairs. If identified in the same direction, then the quotient space of this 2-gon would be a sphere, thus a compact surface. If they are identified in the opposite direction, then it would be a projective plane, hence a compact surface.

Now, suppose that we are talking about a 2n-gon. Now, i know that the quotient space of a 2n-gon with sides identified as follows:

[tex]a_1a_1a_2a_2...a_na_n[/tex] is homeomorphic to a connected sum of n projective planes, thus by the above theorem it is compact.

I believe, the proof would be concluded if we could show that any other identification could somehow be transformed into either the connected sum of n projective planes or n tori.

Any suggestions would be greatly appreciated!

Thnx in advance.