In http://en.wikipedia.org/wiki/Universal_coefficient_theorem they use the Universtal coefficient theorem
$0 \rightarrow \mbox{Ext}(H_{i-1}(X, \mathbb{Z}),A)\rightarrow H^i(X,A)\rightarrow\mbox{Hom}(H_i(X, \mathbb{Z}),A)\rightarrow 0$
to determine the cohomology of $RP^n$ with coefficient in the field $\mathbb Z_2$ which gives that: $\forall i = 0 \ldots n , \ H^i (RP^n; \mathbb Z_2) = \mathbb Z_2$
But in Hatcher page 198, he says that with field coefficients cohomology is the dual of homology that is $H^k(X,\mathbb Z_2)=Hom(H_k(X,\mathbb Z_2);\mathbb Z_2)$. I applied this to get
$H^i(RP^n; \mathbb Z_2) = \begin{cases} \mathbb Z_2 & i = 0 \mbox{ and}, 0
I thought it was a problem in the use of the Universal coefficients theorem taking $\mathbb Z_2$ as a group not as a field but in Hatcher page 199 he says that when the field is $\mathbb Z_p$ or $\mathbb Q$ it does not matter! I'm confused.. thanks for help