Formula for Euler characteristic for quotient space of a CW complex

Question

I know that there is formula for Euler characteristic:

$$\chi(A\cup B)=\chi(A)+\chi(B)-\chi(A\cap B)$$

Is there any formula that links between (for CW complex) some complex, subcomplex and quotient complex ($\chi(X)$,$\chi(A)$,$\chi(X/A)$)?

I appreciate any help.

Answer 1

$\chi(X)=\chi(A)+\chi(X,A)$, see A. Dold, Lectures on Algebraic Topology, Chapt.V, Prop.5.7.

See also the remark of studiosus below.