Show that $X$ is homeomorphic to exactly one of the spaces in the following list: $S^2, P^2, K, T_n, T_n\#P^2,T_n\#K, n > 0$ Where X is a space obtained by pasting the edges of a polygonal region together in pairs. 
Alternatively: Show that X is homeomorphic to exactly one of the spaces in the following list: $S^2,T_n, P^2, K_m, P^2\#K_m$, where $K_m$ is the m-fold connected sum of $K$(Klein bottle) with itself and $M \geq 0$.
We have the classification theorem: If X is the space obtained from a polygonal region in the plane by pasting its edges together in pairs. Then X is homeomorphic to either $S^2, T_n$ or $P_m$. Where $P_m$ is the m-fold projective plane.
It seems I have to show that one of the things on the list that is not $S^2$ or $T_n$ is homeomorphic to $P_m$? The likely candidates seem to be, for the list in the title: $T_n\#P^2$. For the second list: $P^2\#K_m$. But I'm not sure if these are correct and how to formally show that those are homeomorphic to $P_m$
 A: Rather than solving the problem for you, I collect here all relevant facts. Given these facts, actually solving it should be fairly straightforward.
Convention. Write $T_0$ for the $2$-sphere.
Classification Theorem. Suppose $X$ is a compact surface.


*

*If $X$ is orientable, then $X \cong T_n$ for some integer $n \geq 0$.

*If $X$ is non-orientable, then $X \cong P_n$ for some integer $n \geq 1$.



Definition. Letting $\chi$ denote the Euler characteristic function, define $\psi$ as the function on compact surfaces given as
  follows. $$\psi(X) = 2 - \chi(X).$$

Proposition 0.


*

*$\psi(X \,\#\,Y) = \psi(X)+\psi(Y),$ for all compact surfaces $X$ and $Y$.

*$\psi(T_0) = 0$


Proposition 1. $\psi(T) = 2, \psi(P) = 1$
Corollary. $\psi(T_n) = 2n, \psi(P_n) = n$
Proposition 2. (Dyck's theorem.) $T \,\#\, P = P \,\#\,P\,\#\,P$
Proposition 3. If $K$ is the Klein bottle, then:
$$K \cong P \,\#\, P$$
A: I think all you need to know is that the sum of three projective planes is homeomorphic to the sum of a torus and a projective plane. 
