quality:corruption
Inclusion–exclusion principle

If M and N are any two topological spaces, then the Euler characteristic of their disjoint union is the sum of their Euler characteristics, since homology is additive under disjoint union:

More generally, if M and N are subspaces of a larger space X, then so are their union and intersection. In some cases, the Euler characteristic obeys a version of the inclusion–exclusion principle:

This is true in the following cases:

In general, the inclusion–exclusion principle is false. A counterexample is given by taking X to be the real line, M a subset consisting of one point and N the complement of M.