If $M$ and $N$ are sufficiently nice subspaces of some topological space $X$ then their Euler characteristics obey an inclusion-exclusion principle: \begin{equation} \chi(M) + \chi(N) = \chi(M\cup N) - \chi(M\cap N), \end{equation} where $\chi$ denotes the Euler characteristic. I have seen it written, for example on Wikipedia, that in fact this is true in greater generality for the Euler characteristic with compact supports. The formula above holds for any $M$ and $N$, provided $X$ is locally compact and the compactly supported Euler characteristic is used instead of the standard Euler characteristic. Why is this? What would be a good reference for this fact?

If $M$ and $N$ are sufficiently nice subspaces of some topological space $X$ then their Euler characteristics obey an inclusion-exclusion principle: \begin{equation} \chi(M) + \chi(N) = \chi(M\cup N) + \chi(M\cap N), \end{equation} where $\chi$ denotes the Euler characteristic. I have seen it written, for example on Wikipedia, that in fact this is true in greater generality for the Euler characteristic with compact supports. The formula above holds for any $M$ and $N$, provided $X$ is locally compact and the compactly supported Euler characteristic is used instead of the standard Euler characteristic. Why is this? What would be a good reference for this fact?