Editing Euler characteristic (section)

===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:

:<math>\chi(M \sqcup N) = \chi(M) + \chi(N).</math>

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]]:

:<math>\chi(M \cup N) = \chi(M) + \chi(N) - \chi(M \cap N).</math>

This is true in the following cases:

*if ''M'' and ''N'' are an [[excisive couple]]. In particular, if the [[interior (topology)|interiors]] of ''M'' and ''N'' inside the union still cover the union.<ref>Edwin Spanier: Algebraic Topology, Springer 1966, p. 205.</ref>
*if ''X'' is a [[locally compact space]], and one uses Euler characteristics with [[compact space|compact]] [[support (mathematics)|supports]], no assumptions on ''M'' or ''N'' are needed.
*if ''X'' is a [[topologically stratified space|stratified space]] all of whose strata are even-dimensional, the inclusion–exclusion principle holds if ''M'' and ''N'' are unions of strata.  This applies in particular if ''M'' and ''N'' are subvarieties of a [[complex number|complex]] [[algebraic variety]].<ref>William Fulton: Introduction to toric varieties, 1993, Princeton University Press, p. 141.</ref>

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 (set theory)|complement]] of ''M''.