If a finite CW complex $X$ is the union of sub complexes $A$ and $B$, show that $\chi(X)=\chi (A)+\chi (B)-\chi (A \cap B)$.
some how I can imagine what is happening,it is counting numbers of all kind of holes and calculate the sum of them with respect to the signs,and it subtracts the holes that counted twice one in $A$ and one in $B$.
I must make it rigid to connect my imagination to the theory,I don't know how should I start.please give me guidance or any note thank you very much.