Given a pushout $P=B\cup_AC$ which we represent as a commutative diagram
$ \begin{matrix} A & \stackrel{f}{\rightarrow} & B\\ \downarrow{g} & &\downarrow{k} \\ C &\stackrel{h}{\rightarrow} & P \end{matrix} $ the euler characteristic is given by $\chi(P)=\chi(C)+\chi(B)-\chi(A)$.
Do we have a similar situation when a space is constructed from a pullback, i mean what can be said of $\chi(X)$ when a space $X$ is given by a pullback : $ \begin{matrix} X&\stackrel{f}{\rightarrow}&Y\\ \downarrow{g}&&\downarrow{k}\\ Z&\stackrel{h}{\rightarrow}&T \end{matrix}$