Euler characteristic obeys inclusion-exclusion for finite CW subcomplexes.Suggest a slogan
Let \(X\) be a finite CW complex and \(U, V\subseteq X\) subcomplexes with \(X = U\cup V\). Then \[ \chi(X) = \chi(U) + \chi(V) - \chi (U\cap V) .\]
This follows by counting cells, since the Euler characteristic of a finite CW complex is the alternating sum of its numbers of cells [@Hat02].